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DESIGN FOR A BRAIN 



DESIGN FOR A BRAIN 



W. ROSS ASHBY 

M.A., M.D. (Cantab), D.P.M. 

Director of Research 
Barnwood House, Gloucester 






REPRINTED 
(with corrections) 




NEW YORK 
JOHN WILEY & SONS INC, 

440 FOURTH AVENUE 
1954 



First Published . . . 1952 

Reprinted (with corrections) . 1954 



Printed in Great Britain by Butler & Tanner Ltd., Frome and London 



Summary and Preface* 

The book is not a treatise on all cerebral mechanisms but an 
attempt to solve a specific problem : the origin of the nervous 
system's unique ability to produce adaptive behaviour. The 
work has as basis the fact that the nervous system behaves 
adaptively and the hypothesis that it is essentially mechanistic ; 
it proceeds on the assumption that these two data are not irre- 
concilable. It attempts to deduce from the observed facts what 
sort of a mechanism it must be that behaves so differently from 
any machine made so far. Many other workers have proposed 
theories on the subject, but they have usually left open the 
question whether some different theory might not fit the facts 
equally well. I have attempted to deduce what is necessary, 
what properties the nervous system must have if it is to behave 
at once mechanistically and adaptively. 

Proceeding in this way I have deduced that any system which 
shows adaptation must (1) contain many variables that behave 
as step-functions, (2) contain many that behave as part-functions, 
and (3) be assembled largely at random, so that its details are 
determined not individually but statistically. The last require- 
ment may seem surprising : man-made machines are usually 
built to an exact specification, so we might expect a machine 
assembled at random to be wholly chaotic. But it appears that 
this is not so. Such a system has a fundamental tendency, shown 
most clearly when its variables are numerous, to so arrange its 
internal pattern of action that, in relation to its environment, it 
becomes stable. If the system were inert this would mean little ; 
but in a system as active and complex as the brain, it implies that 
the system will be self-preserving through active and complex 
behaviour. 

The work may also be regarded as amplifying the view that 
the nervous system is not only sensitive but ' delicate ' : that its 
encounters with the environment mark it readily, extensively, 
and permanently, with traces distributed according to the 
4 accidents ' of the encounter. Such a distribution might be 
expected to produce a merely chaotic alteration in the nervous 
system's behaviour, but this is not so : as the encounters multiply 
there is a fundamental tendency for the system's adaptation to 
improve, for the traces tend to such a distribution as will make 
its behaviour adaptive in the subsequent encounters. 

* The summary is too brief to be accurate ; the full text should be con- 
sulted for the necessary qualifications. 

V 



SUMMARY AND PREFACE 

The work also in a sense develops a theory of the ' natural 
selection ' of behaviour-patterns. Just as, in the species, the 
truism that the dead cannot breed implies that there is a funda- 
mental tendency for the successful to replace the unsuccessful, 
so in the nervous system does the truism that the unstable tends 
to destroy itself imply that there is a fundamental tendency for 
the stable to replace the unstable. Just as the gene-pattern, in 
its encounters with the environment, tends towards ever better 
adaptation of the inherited form and function, so does a system 
of step- and part-functions tend towards ever better adaptation 
of learned behaviour. 

These remarks give an impressionist picture of the work's 
nature ; but a description in these terms is not well suited to 
systematic exposition. The book therefore presents the evidence 
in rather different order. The first five chapters are concerned 
with foundations : with the accurate definition of concepts, with 
basic methods, and especially with the establishing of exact 
equivalences between the necessary physical, physiological, and 
psychological concepts. After the development of more advanced 
concepts in the next two chapters, the exposition arrives at its 
point : the principle of ultrastability, which in Chapter 8 is defined 
and described. The next two chapters apply it to the nervous 
system and show how it explains the organism's basic power of 
adaptation. The remainder of the book studies its developments : 
Chapters 11 to 13 show the inadequacy of the principle in systems 
that lack part-functions, Chapters 14 to 16 develop the properties 
of systems that contain them, and Chapters 17 and 18 offer 
evidence that the principle's power to develop adaptation is 
unlimited. 

The thesis is stated twice : at first in plain words and then in 
mathematical form. Having experienced the confusion that 
tends to arise whenever we try to relate cerebral mechanisms to 
psychological phenomena, I made it my aim to accept nothing 
that could not be stated in mathematical form, for only in this 
language can one be sure, during one's progress, that one is not 
unconsciously changing the meaning of terms, or adding assump- 
tions, or otherwise drifting towards confusion. The aim proved 
achievable. The concepts of organisation, behaviour, change of 
behaviour, part, whole, dynamic system, co-ordination, etc. — 
notoriously elusive but essential — were successfully given rigorous 
definition and welded into a coherent whole. But the rigour 
and coherence depended on the mathematical form, which is not 
read with ease by everybody. As the basic thesis, however, 
rested on essentially common-sense reasoning, I have been able to 
divide the account into two parts. The main account (Chapters 
1-18) is non-mathematical and is complete in itself. The Appen- 
dix (Chapters 19-24) contains the definitive theory in mathema- 

vi 



SUMMARY AND PREFACE 

tical form. So far as is possible, the main account and the 
Appendix have been written in parallel to facilitate cross-reference. 

Since the reader will probably need cross-reference frequently, 
the chapters have been subdivided into sections. These are 
indicated thus : S. 4/5, which means Chapter 4's fifth section. 
Each figure and table is numbered within its own section : Fig. 
4/5/2 is the second figure in S. 4/5. Section-numbers are given 
at the top of every page, so finding a section or a figure should be 
as simple and direct as finding a page. 

Figs. 8/8/1 and 8/8/2 are reproduced by permission of the 
Editor of Electronic Engineering. 

It is a pleasure to be able to express my indebtedness to the 
Governors of Barnwood House and to Dr. G. W. T. H. Fleming 
for their generous support during the prosecution of the work, 
and to Professor F. L. Golla and Dr. W. Grey Walter for much 
helpful criticism. 

W. ROSS ASHBY 



vn 



Contents 

CHAPTER PAGE 

Summary and Preface v 

1 The Problem 1 

2 Dynamic Systems 13 

3 The Animal as Machine 29 

4 Stability 43 

5 Adaptation as Stability 57 

6 Parameters 72 

7 Step-functions 80 

8 The Ultrastable System 90 

9 Ultras tability in the Living Organism 103 

10 Step-functions in the Living Organism 125 

11 Fully Connected Systems 130 

12 Iterated Systems 139 

13 Disturbed Systems and Habituation 144 

14 Constancy and Independence 153 

15 Dispersion 166 

16 The Multistage System 171 

17 Serial Adaptation 179 

18 Interaction between Adaptations 190 

APPENDIX 

19 The Absolute System 203 

20 Stability 216 

21 Parameters 226 

22 Step-functions 232 

23 The Ultrastable System 237 

24 Constancy and Independence 241 

Index 255 



ix 



CHAPTER 1 

The Problem 



1/1. How does the brain produce adaptive behaviour ? In 
attempting to answer the question, scientists have discovered two 
sets of facts and have had some difficulty in reconciling them. 
The physiologists have shown in a variety of ways how closely the 
brain resembles a machine : in its dependence on chemical 
reactions, in its dependence on the integrity of anatomical paths, 
and in many other ways. At the same time the psychologists 
have established beyond doubt that the living organism, whether 
human or lower, can produce behaviour of the type called 4 pur- 
poseful ' or ' intelligent ' or ' adaptive ' ; for though these words 
are difficult to define with precision, no one doubts that they 
refer to a real characteristic of behaviour. These two character- 
istics of the brain's behaviour have proved difficult to reconcile, 
and some workers have gone so far as to declare them incom- 
patible. 

Such a point of view will not be taken here. I hope to show 
that a system can be both mechanistic in nature and yet 
produce behaviour that is adaptive. I hope to show that the 
essential difference between the brain and any machine yet made 
is that the brain makes extensive use of a principle hitherto 
little used in machines. I hope to show that by the use of this 
principle a machine's behaviour may be made as adaptive as we 
please, and that the principle may he capable of explaining even 
the adaptiveness of Man. 

But first we must examine more closely the nature of the 
problem, and this will be commenced in this chapter. The suc- 
ceeding chapters will develop more accurate concepts, and when 
we can state the problem with precision we shall be not far from 
its solution. 

1 



1/2 DESIGN FOR A BRAIN 

Behaviour, reflex and learned 

1/2. The activities of the nervous system may be divided more 
or less distinctly into two types. The dichotomy is perhaps an 
over-simplification, but it will be sufficient for our purpose. 

The first type is reflex behaviour. It is inborn, it is genetically 
determined in detail, it is a product, in the vertebrates, chiefly 
of centres in the spinal cord and in the base of the brain, and it is 
not appreciably modified by individual experience. The second 
type is learned behaviour. It is not inborn, it is not genetically 
determined in detail (more fully discussed in S. 1/9), it is a product 
chiefly of the cerebral cortex, and it is modified markedly by the 
organism's individual experiences. 

1/3. With the first or reflex type of behaviour we shall not be 
concerned. We assume that each reflex is produced by some 
neural mechanism whose physico-chemical nature results inevit- 
ably in the characteristic form of behaviour, that this mechanism 
is developed under the control of the gene-pattern and is inborn, 
and that the pattern of behaviour produced by the mechanism is 
usually adapted to the animal's environment because natural 
selection has long since eliminated all non-adapted variations. 
For example, the complex activity of ' coughing ' is assumed to 
be due to a special mechanism in the nervous system, inborn and 
developed by the action of the gene-pattern, and adapted and 
perfected by the fact that an animal who is less able to clear its 
trachea of obstruction has a smaller chance of survival. 

Although the mechanisms underlying these reflex activities are 
often difficult to study physiologically and although few are known 
in all their details, yet it is widely held among physiologists that 
no difficulty of principle is involved. Such behaviour and such 
mechanisms will not therefore be considered further. 

1/4. It is with the second type of behaviour that we are con- 
cerned : the behaviour that is not inborn but learned. Examples 
of such reactions exist in abundance, and any small selection 
must seem paltry. Yet I must say what I mean, if only to give 
the critic a definite target for attack. Several examples will 
therefore be given. 

A dog selected at random for an experiment with a conditioned 

2 



THE PROBLEM 1/5 

reflex can be made at will to react to the sound of a bell either 
with or without salivation. Further, once trained to react in 
one way it may, with little difficulty, be trained to react later in 
the opposite way. The salivary response to the sound of a bell 
cannot, therefore, be due to a mechanism of fixed properties. 

A rat selected at random for an experiment in maze-running 
can be taught to run either to right or left by the use of an appro- 
priately shaped maze. Further, once trained to turn to one side 
it can be trained later to turn to the other. 

A kitten approaching a fire for the first time is unpredictable 
in its first reactions. The kitten may walk almost into it, or 
may spit at it, or may dab at it with a paw, or may try to sniff 
at it, or may crouch and ' stalk ' it. The initial way of behaving 
is not, therefore, determined by the animal's species. 

Perhaps the most striking evidence that animals, after training, 
can produce behaviour which cannot possibly have been inborn 
is provided by the circus. A seal balances a ball on its nose for 
minutes at a time ; one bear rides a bicycle, and another walks 
on roller skates. It would be ridiculous to suppose that these 
reactions are due to mechanisms both inborn and specially per- 
fected for these tricks. 

Man himself provides, of course, the most abundant variety of 
learned reactions : but only one example will be given here. If 
one is looking down a compound microscope and finds that the 
object is not central but to the right, one brings the object to 
the centre by pushing the slide still farther to the right. The 
relation between muscular action and consequent visual change 
is the reverse of the usual. The student's initial bewilderment 
and clumsiness demonstrate that there is no neural mechanism 
inborn and ready for the reversed relation. But after a few days 
co-ordination develops. 

These examples, and all the facts of which they are representa- 
tive, show that the nervous system is able to develop ways of 
behaving which are not inborn and are not specified in detail 
by the gene-pattern. 

1/5. Learned behaviour has many characteristics, but we shall 
be concerned chiefly with one : when animals and children learn, 
not only does their behaviour change, but it changes usually for 
the better. The full meaning of i better ' will be discussed in 

3 



1/5 DESIGN FOR A BRAIN 

Chapter 5, but in the simpler cases the improvement is obvious 
enough. ' The burned child dreads the fire ' : after the experi- 
ence the child's behaviour towards the fire is not only changed, 
but is changed to a behaviour which gives a lessened chance of 
its being burned again. We would at once recognise as abnormal 
any child who used its newly acquired knowledge so as to get 
to the flames more quickly. 

To demonstrate that learning usually changes behaviour from a 
less to a more beneficial, i.e. survival-promoting, form would 
need a discussion far exceeding the space available. But in this 
introduction no exhaustive survey is needed. I require only 
sufficient illustration to make the meaning clear. For this pur- 
pose the previous examples will be examined seriatim. 

When a conditioned reflex is established by the giving of food 
or acid, the amount of salivation changes from less to more. And 
the change benefits the animal either by providing normal lubri- 
cation for chewing or by providing water to dilute and flush away 
the irritant. When a rat in a maze has changed its behaviour so 
that it goes directly to the food at the other end, the new behaviour 
is better than the old because it leads more quickly to the animal's 
hunger being satisfied. The kitten's behaviour in the presence of 
a fire changes from being such as may cause injury by burning to 
an accurately adjusted placing of the body so that the cat's body 
is warmed by the fire neither too much nor too little. The circus 
animals' behaviour changes from some random form to one deter- 
mined by the trainer, who applied punishments and rewards. 
The animals' later behaviour is such as has decreased the punish- 
ments or increased the rewards. In Man, the proposition that 
behaviour usually changes for the better with learning would 
need extensive discussion. But in the example of the finger 
movements and the compound microscope, the later movements, 
which bring the desired object directly to the centre of the field, 
are clearly better than the earlier movements, which were dis- 
orderly and ineffective. 

Our problem may now be stated in preliminary form : what 
cerebral changes occur during the learning process, and why does 
the behaviour usually change for the better ? What type of 
mechanistic process could show the same property ? 

But before the solution is attempted we must first glance at the 
peculiar difficulties which will be encountered. 

4 



THE PROBLEM 1/7 

1/6. The nervous system is well provided with means for action. 
Glucose, oxygen, and other metabolites are brought to it by the 
blood so that free energy is available abundantly. The nerve 
cells composing the system are not only themselves exquisitely 
sensitive, but are provided, at the sense organs, with devices of 
even higher sensitivity. Each nerve cell, by its ramifications, 
enables a single impulse to become many impulses, each of which 
is as active as the single impulse from which it originated. And 
by their control of the muscles, the nerve cells can rouse to 
activity engines of high mechanical power. The nervous system, 
then, possesses almost unlimited potentialities for action. But 
do these potentialities solve our problem ? It seems not. We 
are concerned primarily with the question why, during learning, 
behaviour changes for the better : and this question is not 
answered by the fact that a given behaviour can change to one 
of lesser or greater activity. The examples given in S. 1/5, 
when examined for the energy changes before and after learning, 
show that the question of the quantity of activity is usually 
irrelevant. 

But the evidence against regarding mere activity as sufficient 
for a solution is even stronger : often an increase in the amount of 
activity is not so much irrelevant as positively harmful. 

If a dynamic system is allowed to proceed to vigorous action 
without special precautions, the activity will usually lead to the 
destruction of the system itself. A motor car with its tank full 
of petrol may be set into motion, but if it is released with no driver 
its activity, far from being beneficial, will probably cause the 
motor car to destroy itself more quickly than if it had remained 
inactive. The theme is discussed more thoroughly in S. 20/12 ; 
here it may be noted that activity, if inco-ordinated, tends merely 
to the system's destruction. How then is the brain to achieve 
success if its potentialities for action are partly potentialities for 
self-destruction ? 



The relation of part to part 

1/7. It was decided in S. 1/5 that after the learning process the 
behaviour is usually better adapted than before. We ask, there- 
fore, what property must be possessed by the neurons, or by the 
parts of a mechanical ' brain ', so that the manifestation by 

5 B 



1/8 DESIGN FOR A BRAIN 

the neuron of this property shall result in the whole animal's 
behaviour being improved. 

Even if we allow the neuron all the properties of a living 
organism, it is still insufficiently provided. For the improvement 
in the animal's behaviour is often an improvement in relation to 
entities which have no counterpart in the life of a neuron. Thus 
when a dog, given food in an experiment on conditioned reflexes, 
learns to salivate, the behaviour improves because the saliva 
provides a lubricant for chewing. But in the neuron's existence, 
since all its food arrives in solution, neither ' chewing ' nor ' lubri- 
cant ' can have any direct relevance or meaning. Again, a rat 
learns to run through a maze without mistakes ; yet the learning 
has involved neurons which are firmly supported in a close mesh 
of glial fibres and never move in their lives. 

Finally, consider an engine-driver who has just seen a signal 
and whose hand is on the throttle. If the light is red, the 
excitation from the retina must be transmitted through the 
nervous system so that the cells in the motor cortex send impulses 
down to those muscles whose activity makes the throttle close. 
If the light is green, the excitation from the retina must be 
transmitted through the nervous system so that the cells in the 
motor cortex make the throttle open. And the transmission is 
to be handled, and the safety of the train guaranteed, by neurons 
which can form no conception of ' red ', ' green ', ' train ', ' signal ', 
or ' accident ' ! Yet the system works. 

1/8. In some cases there may be a simple mechanism which 
uses the method that a red light activates a chain of nerve-cells 
leading to the muscles which close the throttle while a green light 
activates another chain of nerve-cells leading to the muscles which 
make it open. In this way the effect of the colour of the signal 
might be transmitted through the nervous system in the appro- 
priate way. 

The simplicity of the arrangement is due to the fact that we 
are supposing that the two reactions are using two completely 
separate and independent mechanisms. This separation may well 
occur in the simpler reactions, but it is insufficient to explain the 
events of the more complex reactions. In most cases the ' correct ' 
and the ' incorrect ' neural activities are alike composed of excita- 
tions, of inhibitions, and of other changes that are all physiological, 

6 



THE PROBLEM 1/8 

so that the correctness is determined not by the process itself but 
by the relations which it bears to the other processes. 

This dependence of the ' correctness ' of what is happening at 
one point in the nervous system on what is happening at other 
points would be shown if the engine-driver were to move over to 
the other side of the cab. For if previously a flexion of the elbow 
had closed the throttle, the same action will now open it ; and 
what was the correct pairing of red and green to push and pull 
must now be reversed. So the local action in the nervous system 
can no longer be regarded as 4 correct ' or ' incorrect ', and the 
first simple solution breaks down. 

Another example is given by the activity of chewing in so 
far as it involves the tongue and teeth in movements which must 
be related so that the teeth do not bite the tongue. No move- 
ment of the tongue can by itself be regarded as wholly wrong, for 
a movement which may be wrong when the teeth are just meeting 
may be right when they are parting and food is to be driven on 
to their line. Consequently the activities in the neurons which 
control the movement of the tongue cannot be described as either 
4 correct ' or ' incorrect ' : only when these activities are related to 
those of the neurons which control the jaw movements can a cor- 
rectness be determined ; and this property now belongs, not to either 
separately, but only to the activity of the two in combination. 

These considerations reveal the main peculiarity of the problem. 
When the nervous system learns, it undergoes changes which 
result in its behaviour becoming better adapted to the environ- 
ment. The behaviour depends on the activities of the various 
parts whose individual actions compound for better or worse into 
the whole action. Why, in the living brain, do they always 
compound for the better ? 

If we wish to build an artificial brain the parts must be specified 
in their nature and properties. But how can we specify the 
4 correct ' properties for each part if the correctness depends not 
on the behaviour of each part but on its relations to the other 
parts ? Our problem is to get the parts properly co-ordinated. 
The brain does this automatically. What sort of a machine can 
be ^//-co-ordinating ? 

This is our problem. It will be stated with more precision in 
S. 1/12. But before this statement is reached, some minor topics 
must be discussed. 

7 



1/9 DESIGN FOR A BRAIN 

The genetic control of cerebral function 

1/9. The various species of the animal kingdom differ widely 
in their powers of learning : Man's intelligence, for instance, is 
clearly a species-characteristic, for the higher apes, however well 
trained, never show an intelligence equal to that of the average 
human being. Clearly the power of learning is determined to 
some extent by the inherited gene-pattern. In what way does 
the gene-pattern exert its effect on the learning process ? In 
particular, what part does it play in the adjustments of part to 
part which the previous section showed to be fundamental ? 
Does the gene-pattern determine these adjustments in detail ? 

In Man, the genes number about 50,000 and the neurons number 
about 10,000,000,000. The genes are therefore far too few to 
specify every neuronic interconnection. (The possibility that a 
gene may control several phenotypic features is to some extent 
balanced by the fact that a single phenotypic feature may require 
several genes for its determination.) 

But the strongest evidence against the suggestion that the 
genes exert, in the higher animals, a detailed control over the 
adjustments of part to part is provided by the evidence of S. 1/4. 
A dog, for instance, can be made to respond to the sound of a 
bell either with or without salivation, regardless of its particular 
gene-pattern. It is impossible, therefore, to relate the control of 
salivation to the particular genes possessed by the dog. This 
example, and all the other facts of which it is typical, show that 
the effect of the gene-pattern on the details of the learning process 
cannot be direct. 

The effect, then, must be indirect : the genes fix permanently 
certain function-rules, but do not interfere with the function-rules 
in their detailed application to particular situations. Three 
examples of this type of control will be given in order to illustrate 
its nature. 

In the game of chess, the laws (the function rules) are few and 
have been fixed for a century ; but their effects are as numerous 
as the number of positions to which they can be applied. The 
result is that games of chess can differ from one another though 
controlled by constant laws. 

A second example is given by the process of evolution through 
natural selection. Here again the function-rule (the principle of 

8 



THE PROBLEM 1/10 

the survival of the fittest) is fixed, yet its influence has an infinite 
variety when applied to an infinite variety of particular organisms 
in particular environments. 

A final example is given in the body by the process of inflam- 
mation. The function-rules which govern the process are gene- 
tically determined and are constant in one species. Yet these 
rules, when applied to an infinite variety of individual injuries, 
provide an infinite variety in the details of the process at particular 
points and times. 

Our aim is now clear : we must find the function-rules. They 
must be few in number, much fewer than 50,000, and we must 
show that these few function-rules, when applied to an almost 
infinite number of circumstances and to 10,000,000,000 neurons, 
are capable of directing adequately the events in all these circum- 
stances. The function-rules must be fixed, their applications 
flexible. 

(The gene-pattern is discussed again in S. 9/9.) 



Restrictions on the concepts to be used 

1/10. Throughout the book I shall adhere to certain basic 
assumptions and to certain principles of method. 

The nervous system, and living matter in general, will be 
assumed to be identical with all other matter. So no use of any 
1 vital ' property or tendency will be made, and no deus ex machina 
will be invoked. No psychological concept will be used unless 
it can be shown in objective form in non-living systems ; and 
when used it will be considered to refer solely to its objective 
form. Related is the restriction that every concept used must 
be capable of objective demonstration. In the study of man 
this restriction raises formidable difficulties extending from the 
practical to the metaphysical. But as most of the discussion 
will be concerned with the observed behaviour of animals and 
machines, the peculiar difficulties will seldom arise. 

No teleological explanation for behaviour will be used. It 
will be assumed throughout that a machine or an animal behaved 
in a certain way at a certain moment because its physical and 
chemical nature at that moment allowed it no other action. Never 
will we use the explanation that the action is performed because 
it will later be advantageous to the animal. Any such explanation 

9 



1/11 DESIGN FOR A BRAIN 

would, of course, involve a circular argument ; for our purpose 
is to explain the origin of behaviour which appears to be teleo- 
logically directed. 

It will be further assumed that the nervous system, living 
matter, and the matter of the environment are all strictly deter- 
minate : that if on two occasions they are brought to the same 
state, the same behaviour will follow. Since at the atomic level 
of size the assumption is known to be false, the assumption implies 
that the functional units of the nervous system must be sufficiently 
large to be immune to this source of variation. For this there is 
some evidence, since recordings of nervous activity, even of single 
impulses, show no evidence of appreciable thermal noise. But 
we need not prejudge the question. The work to be described 
is an attempt to follow the assumption of determinacy wherever 
it leads. When it leads to obvious error will be time to question 
its validity. 

Consciousness 

1/11. The previous section has demanded that we shall make no 
use of the subjective elements of experience ; and I can antici- 
pate by saying that in fact the book makes no such use. At 
times its rigid adherence to the objective point of view may 
jar on the reader and may expose me to the accusation that I am 
ignoring an essential factor. A few words in explanation may 
save misunderstanding. 

Throughout the book, consciousness and its related subjective 
elements are not used for the simple reason that at no point have I 
found their introduction necessary. This is not surprising, for the 
book deals with only one of the aspects of the mind-body relation, 
and with an aspect — learning — that has long been recognised to 
have no necessary dependence on consciousness. Here is an 
example to illustrate their independence. If a cyclist wishes to 
turn to the left, his first action must be to turn the front wheel 
to the right (otherwise he will fall outwards by centrifugal force). 
Every practised cyclist makes this movement every time he 
turns, yet many cyclists, even after they have made the move- 
ment hundreds of times, are quite unconscious of making it. 
The direct intervention of consciousness is evidently not necessary 
for adaptive learning. 

10 



THE PROBLEM 1/12 

Such an observation, showing that consciousness is sometimes 
not necessary, gives us no right to deduce that consciousness 
does not exist. The truth is quite otherwise, for the fact of the 
existence of consciousness is prior to all other facts. If I perceive 
— am aware of — a chair, I may later be persuaded, by other 
evidence, that the appearance was produced only by a trick of 
lighting ; I may be persuaded that it occurred in a dream, or 
even that it was an hallucination ; but there is no evidence in 
existence that could persuade me that my awareness itself was 
mistaken— that I had not really been aware at all. This know- 
ledge of personal awareness, therefore, is prior to all other forms 
of knowledge. 

If consciousness is the most fundamental fact of all, why is it 
not used in this book ? The answer, in my opinion, is that 
Science deals, and can deal, only with what one man can demonstrate 
to another. Vivid though consciousness may be to its possessor, 
there is as yet no method known by which he can demonstrate his 
experience to another. And until such a method, or its equivalent, 
is found, the facts of consciousness cannot be used in scientific 
method. 

The problem 

1/12. It is now time to state the problem. Later, when more 
exact concepts have been developed, it will be possible to state the 
problem more precisely (S. 8/1). 

It will be convenient, throughout the discussion, to have some 
well-known, practical problem to act as type-problem, so that 
general statements can always be referred to it. I select the 
following. When a kitten first approaches a fire its reactions are 
unpredictable and usually inappropriate. Later, however, when 
adult, its reactions are different. It approaches the fire and seats 
itself at that place where the heat is moderate. If the fire burns 
low, it moves nearer. If a hot coal falls out, it jumps away. 
I might have taken as type-problem some experiment published 
by a psychological laboratory, but the present example has 
several advantages. It is well known ; it is representative of a 
wide class of important phenomena ; and it is not likely to be 
called in question by the discovery of some small technical flaw. 

With this as specific example, we may state the problem 

11 



1/12 DESIGN FOR A BRAIN 

generally. We commence with the concepts that the organism 
is mechanistic in action, that it is composed of parts, and that 
the behaviour of the whole is the outcome of the compounded 
actions of the parts. Organisms change their behaviour by 
learning, and change it so that the later behaviour is better 
adapted to their environment than the earlier. Our problem is, 
first, to identify the nature of the change which shows as learning, 
and secondly, to find why such changes should tend to cause 
better adaptation for the whole organism. 



12 



CHAPTER 2 

Dynamic Systems 

2/1. In the previous chapter we have repeatedly used the con- 
cepts of a system, of parts in a whole, of the system's behaviour, 
and of its changes of behaviour. These concepts are fundamental 
and must be properly defined. Accurate definition at this stage 
is of the highest importance, for any vagueness here will infect 
all the subsequent discussion ; and as we shall have to enter the 
realm where the physical and the psychological meet, a realm 
where the experience of centuries has found innumerable possi- 
bilities of confusion, we shall have to proceed with unusual 
caution. 

We start by assuming that we have before us some dynamic 
system, i.e. something that may change with time. We wish to 
study it. It will be referred to as the ' machine ', but the word 
must be understood in the widest possible sense, for no restriction 
is implied other than that it should be objective. 

2/2. As we shall be more concerned in this chapter with prin- 
ciples than with practice, we shall be concerned chiefly with 
constructing a method. When constructed, it must satisfy these 
axiomatic demands: — (1) Its procedure for obtaining informa- 
tion must be wholly objective. (2) It must obtain its information 
solely from the ' machine ', no other source being permitted. 
(3) It must be applicable, at least in principle, to all material 
4 machines ', whether animate or inanimate. (4) It must be 
precisely defined. 

The actual form developed may appear to the practical worker 
to be clumsy and inferior to methods already in use ; it probably 
is. But it is not intended to compete with the many specialised 
methods already in use. Such methods are usually adapted to a 
particular class of dynamic systems : electronic circuits, rats in 
mazes, solutions of reacting chemical substances, automatic 
pilots, or heart-lung preparations. The method proposed here 

13 



2/3 DESIGN FOR A BRAIN 

must have the peculiarity that it is applicable to all ; it must, 
so to speak, specialise in generality. 



Variable and system 

2/3. The first step is to record the behaviours of the machine's 
individual parts. To do this we identify any number of suitable 
variables. A variable is a measurable quantity which at every 
instant has a definite numerical value. A ' grandfather ' clock, 
for instance, might provide the following variables : — the angular 
deviation of the pendulum from the vertical ; the angular velocity 
with which the pendulum is moving ; the angular position of a 
particular cog-wheel ; the height of a driving weight ; the 
reading of the minute-hand on the scale ; and the length of the 
pendulum. If there is any doubt whether a particular quantity 
may be admitted as a 4 variable ' I shall use the criterion whether 
it can be represented by a pointer on a dial. I shall, in fact, 
assume that such representation is always used : that the 
experimenter is observing not the parts of the real ' machine ' 
directly but the dials on which the variables are displayed, as 
an engineer watches a control panel. 

Only in this way can we be sure of what sources of information 
are used by the experimenter. Ordinarily, when an experimenter 
examines a machine he makes full use of knowledge ' borrowed ' 
from past experience. If he sees two cogs enmeshed he knows 
that their two rotations will not be independent, even though he 
does not actually see them rotate. This knowledge comes from 
previous experiences in which the mutual relations of similar 
pairs have been tested and observed directly. Such borrowed 
knowledge is, of course, extremely useful, and every skilled 
experimenter brings a great store of it to every experiment. 
Nevertheless, it must be excluded from any fundamental method, 
if only because it is not sufficiently reliable : the unexpected 
sometimes happens ; and the only way to be certain of the rela- 
tion between two parts in a new ' machine ' is to test the rela- 
tionship directly. 

All the quantities used in physics, chemistry, biology, physio- 
logy, and objective psychology, are variables in the defined sense. 
Thus, the position of a limb can be specified numerically by co- 
ordinates of position, and movement of the limb can move a pointer 

14 



DYNAMIC SYSTEMS 2/5 

on a dial. Temperature at a point can be specified numerically 
and can be recorded on a dial. Pressure, angle, electric potential, 
volume, velocity, torque, power, mass, viscosity, humidity, sur- 
face tension, osmotic pressure, specific gravity, and time itself, 
to mention only a few, can all be specified numerically and 
recorded on dials. Eddington's statement on the subject is 
explicit : ' The whole subject matter of exact science consists 
of pointer readings and similar indications.' ' Whatever quan- 
tity we say we are " observing ", the actual procedure nearly 
always ends in reading the position of some kind of indicator on 
a graduated scale or its equivalent.' 

Whether the restriction to dial-readings is justifiable with 
living subjects will be discussed in S. 3/4. 

One minor point should be noticed as it will be needed later. 
The absence of an entity can always be converted to a reading on 
a scale simply by considering the entity to be present but in 
zero degree. Thus, ' still air ' can be treated as a wind blowing at 
m.p.h. ; c darkness ' can be treated as an illumination of foot- 
candles ; and the giving of a drug can be represented by indicating 
that its concentration in the tissues has risen from its usual value 
of per cent. 

2/4. A system is any arbitrarily selected set of variables. It is 
a list nominated by the experimenter, and is quite different from 
the real ' machine '. 

At this stage no naturalness of association is implied, and the 
selection is arbitrary. (' Naturalness ' is discussed in S. 2/14.) 

The variable ' time ' will always be used, so the dials will 
always include a clock. But the status of ' time ' in the method 
is unique, so it is better segregated. I therefore add the qualifi- 
cation that ' time ' is not to be included among the variables of 
a system. 

The Method 

2/5. It will be appreciated that every real i machine ' embodies 
no less than an infinite number of variables, most of which must 
of necessity be ignored. Thus if we were studying the swing of a 
pendulum in relation to its length we would be interested in its 
angular deviation at various times, but we would often ignore 

15 



2/6 



DESIGN FOR A BRAIN 



the chemical composition of the bob, the reflecting power of its 
surface, the electric conductivity of the suspending string, the 
specific gravity of the bob, its shape, the age of the alloy, its 
degree of bacterial contamination, and so on. The list of what 
might be ignored could be extended indefinitely. Faced with 
this infinite number of variables, the experimenter must, and of 
course does, select a definite number for examination — in other 
words, he defines his system. Thus, an experimenter once 
drew up Table 2/5/1. He thereby defined a three- variable 



Time 
(mins.) 


Distance of 
secondary 
coil (cm.) 


Part of skin 
stimulated 


Secretion of 

saliva during 

30 sees. 

(drops) 


• • • 


. . . 


. . . 


• • • 



Table 2/5/1 

system, ready for testing. This experiment being finished, he later 
drew up other tables which included new variables or omitted 
old. By definition these new combinations were new systems. 



2/6. The variables being decided on, the recording apparatus 
is now assumed to be attached to the ' machine ' and the experi- 
menter ready to observe the dials. We must next specify what 
power the experimenter has over the experimental situation. 

It is postulated that the experimenter can control any variable 
he pleases : that he can make any variable take any arbitrary 
value at any arbitrary time. The postulate specifies nothing 
about the methods : it demands only that certain end-results 
are to be available. In most cases the means to be used are 
obvious enough. Take the example of S. 2/3 : an arbitrary 
angular deviation of the pendulum can be enforced at any time 
by direct manipulation ; an arbitrary angular momentum can be 
enforced at any time by an appropriate impulse ; the cog can be 
disconnected and shifted, the driving-weight wound up, the hand 
moved, and the pendulum-bob lowered. 

By repeating the control from instant to instant, the experi- 
menter can force a variable to take any prescribed series of 
values. The postulate, therefore, implies that any variable can 
be forced to follow a prescribed course. 

16 



DYNAMIC SYSTEMS 2/7 

Some systems cannot be forced, for instance the astronomical, 
the meteorological, and those biological systems that are accessible 
to observation but not to experiment. Yet no change is neces- 
sary in principle : the experimenter simply waits until the desired 
set of values occurs during the natural changes of the system, 
and he counts that instant as if it were the instant at which the 
system were started. Thus, though we cannot create a thunder- 
storm, we can observe how swallows react to one simply by 
waiting till one occurs 4 spontaneously '. 

2/7. The 4 machine ' will be studied by applying the primary 
operation, defined thus : The variables are brought to a selected 
state (S. 2/9) by the experimenter's power of control (S. 2/6) ; 
the experimenter decides which variables are to be released and 
which are to be controlled ; at a given moment the selected 
variables are released, so that their behaviour is controlled pri- 
marily by the ' machine ', while the others are forced by the 
experimenter to follow their prescribed courses (which often 
includes their being held constant) ; the behaviours of the vari- 
ables are then recorded. This operation is always used in the 
practical investigation of dynamic systems. Here are some 
examples. 

In chemical dynamics the variables are often the concen- 
trations of substances. Selected concentrations are brought 
together, and from a definite moment are allowed to interact 
while the temperature is held constant. The experimenter records 
the changes which the concentrations undergo with time. 

In a mechanical experiment the variables might be the posi- 
tions and momenta of certain bodies. At a definite instant the 
bodies, started with selected velocities from selected positions, 
are allowed to interact. The experimenter records the changes 
which the velocities and positions undergo with time. 

In studies of the conduction of heat, the variables are the 
temperatures at various places in the heated body. A prescribed 
distribution of temperatures is enforced, and, while the tempera- 
tures of some places are held constant, the variations of the 
other temperatures is observed after the initial moment. 

In physiology, the variables might be the rate of a rabbit's 
heart-beat, the intensity of faradisation applied to the vagus 
nerve, and the concentration of adrenaline in the circulating 

17 



2/8 DESIGN FOR A BRAIN 

blood. The intensity of faradisation will be continuously under 
the experimenter's control. Not improbably it will be kept first 
at zero and then increased. From a given instant the changes 
in the variables will be recorded. 

In experimental psychology, the variables might be 4 the 
number of mistakes made by a rat on a trial in a maze ' and 
4 the amount of cerebral cortex which has been removed sur- 
gically '. The second variable is permanently under the experi- 
menter's control. The experimenter starts the experiment and 
observes how the first variable changes with time while the 
second variable is held constant, or caused to change in some 
prescribed manner. 

While a single primary operation may seem to yield little 
information, the power of the method lies in the fact that the 
experimenter can repeat it with variations, and can relate the 
different responses to the different variations. Thus, after one 
primary operation the next may be varied in any of three ways : 
the system may be changed by the inclusion of new variables 
or by the omission of old ; the initial state may be changed ; 
or the prescribed courses may be changed. By applying these 
variations systematically, in different patterns and groupings, the 
different responses may be interrelated to yield relations. 

By further orderly variations, these relations may be further 
interrelated to yield secondary, or hyper-, relations ; and so on. 
In this way the 4 machine ' may be made to yield more and more 
complex information about its inner organisation. 

2/8. All our concepts will eventually be defined in terms of 
this method. For example, ' environment ' is so defined in S. 3/8, 
' adaptation ' in S. 5/8, and ' stimulus ' in S. 6/6. If any have 
been omitted it is by oversight ; for I hold that this procedure 
is sufficient for their objective definition. 

The Field of a System 

2/9. The state of a system at a given instant is the set of numerical 
values which its variables have at that instant. 

Thus, the six-variable system of S. 2/3 might at some 
instant have the state: —4°, 0-3 radians/sec, 128°, 52 cm., 
42-8 minutes, 88-4 cm. 

18 



DYNAMIC SYSTEMS 2/10 

Two states are equal if and only if the corresponding pairs of 
numerical values are all equal. 

2/10. A line of behaviour is specified by a succession of states 
and the time-intervals between them. The first state in a line of 
behaviour will be called the initial state. Two lines of behaviour 
are equal if all the corresponding pairs of states are equal, and 
if all the corresponding pairs of time-intervals are equal. One 
primary operation yields one line of behaviour. 

There are several ways in which a line of behaviour may be 
recorded. 



m 



B 




Time — *- 

Figure 2/10/1 : Events during an experiment on a conditioned reflex in 
a sheep. Attached to the left foreleg is an electrode by which a shock 
can be administered. Line A records the position of the left forefoot. 
Line B records the sheep's respiratory movements. Line C records 
by a rise (E) the application of the conditioned stimulus : the sound 
of a buzzer. Line D records by a vertical stroke (F) the application of 
the electric shock. (After Liddell et al.) 

The graphical method is exemplified by Figure 2/10/1. The 
four variables form, by definition, the system that is being 
examined. The four simultaneous values at any instant define 
a state. And the succession of states at their particular intervals 
constitute and specify the line of behaviour. The four traces 
specify one line of behaviour. 

Sometimes a line of behaviour can be specified in terms of 
elementary mathematical functions. Such a simplicity is con- 
venient when it occurs, but is rarer in practice than an 

19 



2/11 



DESIGN FOR A BRAIN 



acquaintance with elementary mathematics would suggest. With 
biological material it is rare. 

Another form is the tabular, of which an example is Table 2/10/1. 
Each column defines one state ; the whole table defines one line 
of behaviour (other tables may contain more than one line of 
behaviour). The state at hours is the initial state. 







Time (hours) 







1 


3 


6 


■a 

> 


w 

X 


7-35 
156-7 


7-26 
154-6 


7-28 
1541 


7-29 
151-5 


y 


110-3 


116-7 


118-3 


118-5 


z 


22-2 


15-3 


150 


14-6 



Table 2/10/1 : Blood changes after a dose of ammonium chloride, a; 
= serum pH ; x = serum total base ; y = serum chloride ; z = serum 
bicarbonate ; (the last three in m. eq. per 1.). 

The tabular form has one outstanding advantage : it contains 
the facts and nothing more. Mathematical forms are apt to 
suggest too much : continuity that has not been demonstrated, 
fictitious values between the moments of observation, and an 
accuracy that may not be present. Unless specially mentioned, 
all lines of behaviour will be assumed to be recorded primarily 
in tabular form. 



2/11. The behaviour of a system can also be represented in 
phase-space. By its use simple proofs may be given of many 
statements difficult to prove in the tabular form. 

If a system is composed of two variables, a particular state 
will be specified by two numbers. By ordinary graphic methods, 
the two variables can be represented by axes ; the two values 
will then define a point in the plane. Thus the state in which 
variable x has the value 5 and variable y the value 10 will be 
represented by the point A in Figure 2/11/1. The representative 
point of a state is the 'point whose co-ordinates are respectively 
equal to the values of the variables. By S. 2/4 ' time ' is not to 
be one of the axes. 

20 



IO 




2/12 



IO 



Figure 2/11/1. 



2/12. Suppose next that a system of two variables gave the 
line of behaviour shown in Table 2/12/1. The successive states 
will be graphed, by the method, at positions B, C, and D (Figure 
2/11/1). So the system's behaviour corresponds to a movement 
of the representative point along the line in the phase-space. 

By comparing the Table and the Figure, certain exact corre- 
spondences can be found. Every state of the system corresponds 



Time 


X 


y 



1 
2 
3 


5 

6 

7 
5 


10 
9 

7 
4 



Table 2/12/1. 



uniquely to a point in the plane, and every point in the plane 
(or in some portion of it) to some possible state of the system. 
Further, every line of behaviour of the system corresponds 
uniquely to a line in the plane. If the system has three variables, 
the graph must be in three dimensions, but each state still corre- 
sponds to a point, and each line of behaviour to a line in the 
phase-space. If the number of variables exceeds three, this 
method of graphing is no longer physically possible, but the 

21 c 



2/13 



DESIGN FOR A BRAIN 



correspondence is maintained exactly no matter how numerous 
the variables. 

2/13. A system's field is the phase-space containing all the lines 
of behaviour found by releasing the system from all possible initial 
states. 

In practice, of course, the experimenter would test only a repre- 
sentative sample of the initial states. Some of them will probably 
be tested repeatedly, for the experimenter will usually want to 



15- 



10 



c 
_o 
O 

CP 

O 

£ 



5- 






10 15 

Weight of dog (kg. 



20 



Figure 2/13/1 : Arrow-heads show the direction of movement of the 
representative point ; cross-lines show the positions of the representa- 
tive point at weekly intervals. 

make sure that the system is giving reproducible lines of behaviour. 
Thus in one experiment, in which dogs had been severely bled 
and then placed on a standard diet, their body- weight x and the 
concentration y of haemoglobin in their blood were recorded at 
weekly intervals. This two-variable system, tested from four 
initial states by four primary operations, gave the field shown in 
Figure 2/13/1. Other examples occur frequently later. 

It will be noticed that a field is defined, in accordance with 
S. 2/8, by reference exclusively to the observed values of the 

22 



DYNAMIC SYSTEMS 2/14 

variables and to the results of primary operations on them. It 
is therefore a wholly objective property of the system. 

The concept of ' field ' will be used extensively for two reasons. 
It defines the characteristic behaviour of the system, replacing the 
vague concept of what a system ' does ' or how it 4 behaves ' 
(often describable only in words) by the precise construct of a 

4 field '. From this precision comes the possibility of comparing 
field with field, and therefore of comparing behaviour with 
behaviour. The reader may at first find the method unusual. 
Those who are familiar with the phase-space of mechanics will 
have no difficulty, but other readers may find it helpful if at first, 
whenever the word ' field ' occurs, they substitute for it some 
phrase like ' typical way of behaving \ 

The Natural System 

2/14. In S. 2/4 a system was defined as any arbitrarily selected 
set of variables. The right to arbitrary selection cannot be 
waived, but the time has now come to recognise that both science 
and common sense insist that if a system is to be studied with 
profit its variables must have some naturalness of association. 
But what is ' natural ' ? The problem has inevitably arisen 
after the restriction of S. 2/3, where we repudiated all borrowed 
knowledge. If we restrict our attention to the variables, we find 
that as every real 4 machine ' provides an infinity of variables, 
and as from them we can form another infinity of combinations, 
we need some test to distinguish the natural system from the 
arbitrary. 

One criterion will occur to the practical experimenter at once. 
He knows that if an active and relevant variable is left unobserved 
or uncontrolled the system's behaviour will become capricious, 
not capable of being reproduced at will. This concept may 
readily be made more precise. 

If, on repeatedly applying a primary operation to a system, it 
is found that all the lines of behaviour which follow an initial state 

5 are equal, and if a similar equality occurs after every other initial 
state S', S", . . . , then the system is regular. 

Whether a system is regular or not may be decided by first 
constructing and then examining its field. For if the system is 
regular, from each initial state will go only one line of behaviour, 

23 



2/15 DESIGN FOR A BRAIN 

the subsequent trials merely confirming the first. The concept 
of ' regularity ' thus conforms to the demand of S. 2/8 ; for it 
is definable in terms of the field and is therefore wholly objective. 

The field of a regular system does not change with time. 

If, on testing, a system is found to be not regular, the experi- 
menter is faced with the common problem of what to do with a 
system that will not give reproducible results. Somehow he 
must get regularity. The practical details vary from case to 
case, but in principle the necessity is always the same : he must 
try a new system. This means that new variables must be 
added to the previous set, or, more rarely, some irrelevant variable 
omitted. 

From now on we shall be concerned mostly with regular sys- 
tems. We assume that preliminary investigations have been 
completed and that we have found a system, based on the real 
* machine ', that (1) includes the variables in which we are specially 
interested, and (2) includes sufficient other variables to render 
the whole system regular. 

2/15. For some purposes regularity of the system may be 
sufficient, but more often a further demand is made before the 
system is acceptable to the experimenter : it must be ' absolute \* 
It will be convenient if I first define the concept, leaving the 
discussion of its importance to the next section. 

If, on repeatedly applying primary operations to a system, it is 
found that all the lines of behaviour which follow a state S are equal, 
no matter how the system arrived at S, and if a similar equality 
occurs after every other state S', S", . . . , then the system is 
absolute. 

Consider, for instance, the two- variable system that gave the 
two lines of behaviour shown in Table 2/15/1. 

On the first line of behaviour the state x = 0, y = 2-0 was 
followed after 0-1 seconds by the state x = 0-2, y = 2-1. On 
line 2 the state x = 0, y = 2-0 occurred again ; but after 0*1 
seconds the state became x = 0-1, y = 1*8 and not x = 0-2, 
y == 2*1. As the two lines of behaviour that follow the state 
x = 0, y — 2-0 are not equal, the system is not absolute. 

A well-known example of an absolute system is given by the 

* (O.E.D.) Absolute : existent without relation to any other thing ; self- 
sufficing ; disengaged from all interrupting causes. 

24 



DYNAMIC SYSTEMS 



2/15 



simple pendulum swinging in a vertical plane. It is known that 
the two variables — (x) angle of deviation of the string from 







Time (seconds) 


Line 


Variable 





01 


0-2 


0-3 


1 


X 

y 



20 


0-2 
2-1 


0-4 
2-3 


0-6 
2-6 


2 


X 

y 


- 0-2 
2-4 


- 01 
2-2 



20 


01 
1-8 



Table 2/15/1. 

vertical, (y) angular velocity (or momentum) of the bob — are 
such that, all else being kept constant, their two values at a 
given instant are sufficient to determine the subsequent changes 
of the two variables (Figure 2/15/1.) 




Figure 2/15/1 : Field of a simple pendulum 40 cm. long swinging in a 
vertical plane when g is 981 cm. /sec. 2 , x is the angle of deviation from 
the vertical and y the angular velocity of movement. Cross-strokes 
mark the position of the representative point at each one- tenth second. 
The clockwise direction should be noticed. 



An absolute system is thus ' state-determined ', and this is its 
most important property : the occurrence of a state is sufficient 
to determine the line of behaviour that ensues. The property 

25 




2/16 DESIGN FOR A BRAIN 

is both necessary and sufficient ; so all state-determined systems 
are absolute. We shall use this fact repeatedly. 

The field of an absolute system is characteristic : from every 
point there goes only one line of behaviour whether the point is 
initial on the line or not. The field of the two-variable system 
just mentioned is sketched in Figure 2/15/1 ; through every point 
passes only one line. 

These relations may be made clearer if this field is contrasted 
with one that is regular but not absolute. Figure 2/15/2 shows 

such a field (the system is described 
in S. 19/15). The system's regularity 
would be established if we found that 
the system, started at A, always went 
to A\ and, started at B, always went 
to B\ But such a system is not 
absolute ; for to say that the repre- 
sentative point is leaving C is insuf- 
ficient to define its future line of 
behaviour, which may go to A' or B' . 
Figure 2/15/2 : The field Even if the lines from A and B always 
Figure ijfivi. 8110 ™ in ran to A' and B', the regularity in 

no way restricts what would happen 
if the system were started at C : it might go to D. If 
the system were absolute, the lines CA', CB\ and CD would 
coincide. 

A system's absoluteness is determined by its field ; the property 
is therefore wholly objective. 

An absolute system's field does not change with time. 

2/16. We can now return to the question of what we mean when 
we say that a system's variables have a c natural ' association. 
What we need is not a verbal explanation but a definition, which 
must have these properties : 

(1) it must be in the form of a test, separating all systems 

into two classes ; 

(2) its application must be wholly objective ; 

(3) its result must agree with common sense in typical and 

undisputed cases. 
The third property makes clear that we cannot expect a proposed 
definition to be established by a few lines of verbal argument : 



DYNAMIC SYSTEMS 2/17 

it must be treated as a working hypothesis and used ; only experi- 
ence can show whether it is faulty or sound. 

From here on I shall treat a ' natural ' system as equivalent to 
an i absolute ' system. Various reasons might be given to make 
the equivalence plausible, but they would prove nothing and I 
shall omit them. Much stronger is the evidence in the Appendix. 
There it will be found that the equivalence brings clarity where 
there might be confusion ; and it enables proof to be given to 
propositions which, though clear to physical intuition, cannot 
be proved without it. The equivalence, in short, is indispens- 
able. 

Why the concept is so important can be indicated briefly. 
When working with determinate systems the experimenter always 
assumes that, if he is interested in certain variables, he can find 
a set of variables that (1) includes those variables, and (2) has 
the property that if all is known about the set at one instant the 
behaviour of all the variables will be predictable. The assump- 
tion is implicit in almost all science, but, being fundamental, it 
is seldom mentioned explicitly. Temple, though, refers to ' . . . 
the fundamental assumption of macrophysics that a complete 
knowledge of the present state of a system furnishes sufficient 
data to determine definitely its state at any future time or its 
response to any external influence '. Laplace made the same 
assumption about the whole universe when he stated that, given 
its state at one instant, its future progress should be calculable. 
The definition given above makes this assumption precise and 
gives it in a form ready for use in the later chapters. 

2/17. To conclude, here is an example to illustrate this chapter's 
method. 

Suppose someone constructed two simple pendulums, hung them 
so that they swung independently, and from this l machine ' 
brought to an observation panel the following six variables : 

(v) the angular deviation of the first pendulum 

(w) „ „ „ „ „ second 

(x) the angular momentum of the first pendulum 

{y) 9, » » ,, a second „ 

(z) the brightness of their illumination 

(t) the time. 
The experimenter, knowing nothing of the real 4 machine ', or of 

27 



2/17 DESIGN FOR A BRAIN 

the relations between the five variables, sits at the panel and 
applies the defined method. 

He starts by selecting a system at random, constructs its field 
by S. 2/13, and deduces by S. 2/15 whether it is absolute. He 
then tries another system. It is clear that he will eventually be 
able to state, without using borrowed knowledge, that just three 
systems are absolute : (v, w, x, y), {v, x\ and (w, y). He will add 
that z is unpredictable. He has in fact identified the natural 
relations existing in the ' machine '. He will also, at the end of 
his investigation, be able to write down the differential equations 
governing the systems (S. 19/20). Later, by using the method 
of S. 14/6, he will be able to deduce that the four-variable system 
really consists of two independent parts. 

References 

Eddington, A. S. The nature of the physical world. Cambridge, 1929 ; 

The philosophy of physical science. Cambridge, 1939. 
Liddell, H. S., Anderson, O. D., Kotyuka, E., and Hartman, F. A. 

Effect of extract of adrenal cortex on experimental neurosis in sheep. 

Archives of Neurology and Psychiatry, 34, 973 ; 1935. 
Temple, G. General principles of quantum theory. London. Second edition, 

1942. 



28 



CHAPTER 3 

The Animal as Machine 



3/1. We shall assume at once that the living organism in its 
nature and processes is not essentially different from other matter. 
The truth of the assumption will not be discussed. The chapter 
will therefore deal only with the technique of applying this 
assumption to the complexities of biological systems. 



The numerical specification of behaviour 

3/2. If the method laid down in the previous chapter is to be 
followed, we must first determine to what extent the behaviour 
of an organism is capable of being specified by variables, remem- 
bering that our ultimate test is whether the representation can 
be by dial readings (S. 2/3). 

There can be little doubt that any single quantity observable 
in the living organism can be treated at least in principle as a 
variable. All bodily movements can be specified by co-ordinates. 
All joint movements can be specified by angles. Muscle tensions 
can be specified by their pull in dynes. Muscle movements can 
be specified by co-ordinates based on the bony structure or on 
some fixed external point, and can therefore be recorded numeric- 
ally. A gland can be specified in its activity by its rate of 
secretion. Pulse-rate, blood-pressure, temperature, rate of blood- 
flow, tension of smooth muscle, and a host of other variables can 
be similarly recorded. 

In the nervous system our attempts to observe, measure, and 
record have met great technical difficulties. Nevertheless, much 
has been achieved. The action potential, the essential event in 
the activity of the nervous system, can now be measured and 
recorded. The excitatory and inhibitory states of the centres are 
at the moment not directly recordable, but there is no reason to 
suppose that they will never become so. 

29 



3/3 DESIGN FOR A BRAIN 

3/3. Few would deny that the elementary physico-chemical 
events in the living organism can be treated as variables. But 
some may hesitate before accepting that readings on dials are 
adequate for the description of all significant biological events. 
As the remainder of the book will assume that they are sufficient, 
I must show how the various complexities of biological experience 
can be reduced to this standard form. 

A simple case which may be mentioned first occurs when an 
event is recorded in the form ' strychnine was injected at this 
moment ', or ' a light was switched on ', or ' an electric shock was 
administered '. Such a statement treats only the positive event 
as having existence and ignores the other state as a nullity. It 
can readily be converted to a numerical form suitable for our 
purpose by using the device mentioned in S. 2/3. Such events 
would then be recorded by assuming, in the first case, that the 
animal always had strychnine in its tissues but that at first the 
quantity present was mg. per g. tissue ; in the second case, that 
the light was always on, but that at first it shone with a brightness 
of candlepower ; and in the last case, that an electric potential 
was applied throughout but that at first it had a value of volts. 
Such a method of description cannot be wrong in these cases for 
it defines exactly the same set of objective facts. Its advantage 
from our point of view is that it provides a method which can be 
used uniformly over a wide range of phenomena : the variable is 
always present, merely varying in value. 

But this device does not remove all difficulties. It sometimes 
happens in physiology and psychology that a variable seems to have 
no numerical counter-part. Thus in one experiment two cards, 
one black and one brown, were shown alternately to an animal as 
stimuli. One variable would thus be 4 colour ' and it would have 
two values. The simplest way to specify colour numerically is to 
give the wave-length of its light ; but this method cannot be used 
here, for ' black ' means ' no light ', and ' brown ' does not occur 
in the spectrum. Another example would occur if an electric 
heater were regularly used and if its switch indicated only the 
degrees 4 high ', ' medium ', and i low '. Another example is given 
on many types of electric apparatus by a pilot light which, as a 
variable, takes only the two values ' lit ' and 4 unlit '. More 
complex examples occur frequently in psychological experiments. 
Table 2/5/1, for instance, contains a variable ' part of skin stimu- 

30 



THE ANIMAL AS MACHINE 3/3 

lated ' which, in Pavlov's table, takes only two values : ' usual 
place ' and ' new place '. Even more complicated variables are 
common in Pavlov's experiments. Many a table contains a 
variable 4 stimulus ' which takes such values as ' bubbling water ', 
4 metronome ', ' flashing light '. A similar difficulty occurs when 
an experimenter tests an animal's response to injections of toxins, 
so that there will be a variable ' type of toxin ' which may take 
the two values l Diphtheria type Gravis ' and ' Diphtheria type 
Medius '. And finally the change may involve an extensive 
re-organisation of the whole experimental situation. Such would 
occur if the experimenter, wanting to test the effect of the general 
surroundings, tried the effect of the variable ' situation of the 
experiment ' by giving it alternately the two values 4 in the 
animal house ' and ' in the open air '. Can such variables be 
represented by number ? 

In some of the examples, the variables might possibly be speci- 
fied numerically by a more or less elaborate specification of their 
physical nature. Thus ' part of skin stimulated ' might be 
specified by reference to some system of co-ordinates marked on 
the skin ; and the three intensities of the electric heater might be 
specified by the three values of the watts consumed. But this 
method is hardly possible in the remainder of the cases ; nor is it 
necessary. For numbers can be used cardinally as well as 
ordinally, that is, they may be used as mere labels without any 
reference to their natural order. Such are the numberings of the 
divisions of an army, and of the subscribers on a telephone sys- 
tem ; for the subscriber whose number is, say, 4051 has no 
particular relation to the subscriber whose number is 4052 : the 
number identifies him but does not relate him. 

It may be shown (S. 21/1) that if a variable takes a few values 
which stand in no simple relation to one another, then each value 
may be allotted an arbitrary number ; and provided that the 
numbers are used systematically throughout the experiment, and 
that their use is confined to the experiment, then no confusion 
can arise. Thus the variable 4 situation of the experiment ' 
might be allotted the arbitrary value of 4 1 ' if the experiment 
occurs in the animal house, and 4 2 ' if it occurs in the open air. 

Although ' situation of the experiment ' involves a great number 
of physical variables, the aggregate may justifiably be treated as 
a single variable provided the arrangement of the experiment is 

31 



3/4 DESIGN FOR A BRAIN 

such that the many variables are used throughout as one aggre- 
gate which can take either of two forms. If, however, the 
aggregate were split in the experiment, as would happen if we 
recorded four classes of results : 

(1) in the animal house in summer 

(2) in the animal house in winter 

(3) in the open air in summer 

(4) in the open air in winter 

then we must either allow the variable ' condition of experiment ' 
to take four values, or we could consider the experiment as 
subject to two variables : ' site of experiment ' and ' season of 
year ', each of which takes two values. According to this method, 
what is important is not the material structure of the technical 
devices but the experiment's logical structure. 

3/4. But is the method yet adequate ? Can all the living 
organisms' more subtle qualities be numericised in this way ? On 
this subject there has been much dispute, but we can avoid a part 
of the controversy ; for here we are concerned only with certain 
qualities defined. 

First, we shall be dealing not so much with qualities as with 
behaviour : we shall be dealing, not with what an organism feels 
or thinks, but with what it does. The omission of all subjective 
aspects (S. 1/11) removes from the discussion the most subtle of 
the qualities, while the restriction to overt behaviour makes the 
specification by variable usually easy. Secondly, when the non- 
mathematical reader thinks that there are some complex quantities 
that cannot be adequately represented by number, he is apt 
to think of their representation by a single variable. The use of 
many variables, however, enables systems of considerable com- 
plexity to be treated. Thus a complex system like ' the weather 
over England ', which cannot be treated adequately by a single 
variable, can, by the use of many variables, be treated as ade- 
quately as we please. 

3/5. To illustrate the method for specifying the behaviour of a 
system by variables, two examples will be given. They are of 
little intrinsic interest ; more important is the fact that they 
demonstrate that the method is exact and that it can be extended 
to any extent without loss of precision. 

32 



THE ANIMAL AS MACHINE 3/5 

The first example is from a physiological experiment. A dog 
was subjected to a steady loss of blood at the rate of one per cent 
of its body weight per minute. Recorded are the three variables : 

(a?) rate of blood-flow through the inferior vena cava, 

(y) „ » » » » muscles of a leg, 

(2) „ » ,, » » gut. 

The changes of the variables with time are shown in Figure 3/5/1. 
It will be seen that the changes of the variables show a charac- 
teristic pattern, for the blood-flow through leg and gut falls more 
than that through the inferior vena cava, and this difference is 
characteristic of the body's reaction to haemorrhage. The use 





Figure 3/5/1 : Effect of haemor- 
rhage on the rate of blood-flow 
through : x, the inferior vena cava ; 
y, the muscles of a leg ; and z, the 
gut. (From Rein.) 



Figure 3/5/2 : Phase-space and 
line of behaviour of the data 
shown in Figure 3/5/1. 



of more than one variable has enabled the pattern of the reaction 

to be displayed. 

The changes specify a line of behaviour, shown in Figure 3/5/2. 

Had the line of behaviour pointed in a different direction, the 

change would have corresponded to a change in the pattern of 

the body's reaction to haemorrhage. 

The second example uses certain angles measured from a 

cinematographic record of the activities of a man. His body 

moved forward but was vertical throughout. The four variables 

are : 

(w) angle between the right thigh and the vertical 
[pc) •>■> >» >> leit ,, ,, ,, ,, 

{y) 99 „ » right „ „ „ right tibia 

\~) 5) 5? ?> leit ,, ,, ,, leit ,, 

In w and x the angle is counted positively when the knee comes 

33 



3/6 



DESIGN FOR A BRAIN 



forward : in y and z the angles are measured behind the knee. 
The line of behaviour is specified in Table 3/5/1. The reader can 
easily identify this well-known activity. 







Time (seconds) 







01 


0-2 


0-3 


0-4 


0-5 


OG 


0-7 


0-8 


0) 


XV 


45 


10 


-10 


-20 


-35 





GO 


70 


45 


J2 
C8 


X 


-35 





GO 


70 


45 


10 


-10 


-20 


-35 


f-t 


y 


170 


180 


180 


1G0 


120 


80 


GO 


100 


170 


z 


120 


80 


CO 


100 


170 


180 


180 


160 


120 



Table 3/5/1. 



The organism as system 

3/6. In a physiological experiment the nervous system is usually 
considered to be absolute. That it can be made absolute is 
assumed by every physiologist before the work starts, for he 
assumes that it is subject to the fundamental assumption of 
S. 2/15 : that if every detail within it could be determined, its 
subsequent behaviour would also be determined. Many of the 
specialised techniques such as anaesthesia, spinal transection, 
root section, and the immobilisation of body and head in clamps 
are used to ensure proper isolation of the system — a necessary 
condition for its absoluteness (S. 2/15). So unless there are 
special reasons to the contrary, the nervous system in a physio- 
logical experiment has the properties of an absolute system. 

3/7. Similarly it is usually agreed that an animal undergoing 
experiments on its conditioned reflexes is a physico-chemical 
system such that if we knew every detail we could predict its 
behaviour. Pavlov's insistence on complete isolation was in- 
tended to ensure that this was so. So unless there are special 
reasons to the contrary, the animal in an experiment with con- 
ditioned reflexes has the properties of an absolute system. 



34 



THE ANIMAL AS MACHINE 3/10 

The environment 

3/8. These two examples, however, are mentioned only as 
introduction ; rather we shall be concerned with the nature of the 
free-living organism within a natural environment. 

Given an organism, its environment is defined as those variables 
whose changes affect the organism, and those variables which are 
changed by the organism's behaviour. It is thus defined in a purely 
functional, not a material, sense. It will be treated uniformly 
with our treatment of all variables : we assume it is represent- 
able by dials, is explorable (by the experimenter) by primary 
operations, and is intrinsically determinate. 

Organism and environment 

3/9. The theme of the chapter can now be stated : the free- 
living organism and its environment, taken together, form an 
absolute system. 

The concepts developed in the previous sections now enable us 
to treat both organism and environment by identical methods, 
for the same primary assumptions are made about each. The 
two parts act and re-act on one another (S. 3/11), and are there- 
fore properly regarded as two parts of one system. And since 
we have assumed that the conjoint system is state-determined, 
we may treat the whole as absolute. 

3/10. As example, that the organism and its environment form 
a single absolute system, consider (in so far as the activities of 
balancing are concerned) a bicycle and its rider in normal 
progression. 

First, the forward movement may be eliminated as irrelevant, 
for we could study the properties of this dynamic system equally 
well if the wheels were on some backward-moving band. The 
variables canjbe identified by considering what happens. Suppose 
the rider pulls his right hand backwards : it will change the 
angular position of the front wheel (taking the line of the frame as 
reference). The changed angle of the front wheel will start the 
two points, at which the wheels make contact with the ground, 
moving to the right. (The physical reasons for this movement 
are irrelevant : the fact that the relation is determined is sufficient.) 

35 



3/11 DESIGN FOR A BRAIN 

The rider's centre of gravity being at first unmoved, the line 
vertically downwards from his centre of gravity will strike the 
ground more and more to the left of the line joining the two 
points. As a result he will start to fall to the left. This fall will 
excite nerve-endings in the organs of balance in the ear, impulses 
will pass to the nervous system, and will be switched through it, 
if he is a trained rider, by such a route that they, or the effects 
set up by them, will excite to activity those muscles which push 
the right hand forwards. 

We can now specify the variables which must compose the 
system if it is to be absolute. We must include : the angular 
position of the handlebar, the velocity of lateral movement of the 
two points of contact between wheels and road, the distance 
laterally between the line joining these points and the point 
vertically below the rider's centre of gravity, and the angular 
deviation of the rider from the vertical. These four variables are 
denned by S. 3/8 to be the 4 environment ' of the rider. (Whether 
the fourth variable is allotted to ' rider ' or to i environment ' is 
optional (S. 3/12) ). To make the system absolute, there must be 
added the variables of the nervous system, of the relevant muscles, 
and of the bone and joint positions. 

As a second example, consider a butterfly and a bird in the air, 
the bird chasing the butterfly, and the butterfly evading the bird. 
Both use the air around them. Every movement of the bird 
stimulates the butterfly's eye and this stimulation, acting through 
the butterfly's nervous system, will cause changes in the butter- 
fly's wing movements. These movements act on the enveloping 
air and cause changes in the butterfly's position. A change of 
position immediately changes the excitations in the bird's eye, 
and this leads through its nervous system to changed movements 
of the bird's wings. These act on the air and change the bird's 
position. So the processes go on. The bird has as environment 
the air and the butterfly, while the butterfly has the bird and the 
air. The whole may justifiably be assumed absolute. 

3/11. The organism affects the environment, and the environ- 
ment affects the organism : such a system is said to have 4 feed- 
back ' (S. 4/12). 

The examples of the previous section provide illustration. 
The rider's arm moves the handlebars, causing changes in the 

36 



THE ANIMAL AS MACHINE 3/11 

environment ; and changes in these variables will, through the 
rider's sensory receptors, cause changes in his brain and muscles. 
When bird and butterfly manoeuvre in the air, each manoeuvre 
of one causes reactive changes to occur in the other. 

The same feature is shown by the example of S. 1/12 — the 
type problem of the kitten and the fire. The various stimuli 
from the fire, working through the nervous system, evoke some 
reaction from the kitten's muscles ; equally the kitten's move- 
ments, by altering the position of its body in relation to the fire, 
will cause changes to occur in the pattern of stimuli which falls 
on the kitten's sense-organs. The receptors therefore affect the 
muscles (by effects transmitted through the nervous system), and 
the muscles affect the receptors (by effects transmitted through 
the environment). The action is two-way and the system possesses 
feedback. 

The observation is not new : — 

' In most cases the change which induces a reaction is brought 
about by the organism's own movements. These cause a 
change in the relation of the organism to the environment : 
to these changes the organism reacts. The whole behaviour 
of free-moving organisms is based on the principle that it 
is the movements of the organism that have brought about 
stimulation.' 

(Jennings.) 

' The good player of a quick ball game, the surgeon con- 
ducting an operation, the physician arriving at a clinical 
decision — in each case there is the flow from signals inter- 
preted to action carried out, back to further signals and on 
again to more action, up to the culminating point of the achieve- 
ment of the task '. 

(Bartlett.) 

4 Organism and environment form a whole and must be 
viewed as such.' 

(Starling.) 

It is necessary to point to the existence of feedback in the 
relation between the free-living organism and its environment 
because most physiological experiments are deliberately arranged 
to avoid feedback. Thus, in an experiment with spinal reflexes, 
a stimulus is applied and the resulting movement recorded ; but 
the movement is not allowed to influence the nature or duration 
of the stimulus. The action between stimulus and movement is 
therefore one-way. A similar absence of feedback is enforced 

37 D 



3/11 DESIGN FOR A BRAIN 

in the Pavlovian experiments with conditioned reflexes : the 
stimulus may evoke salivation, but the salivation has no effect 
on the nature or duration of the stimulus. 

Such an absence of feedback is, of course, useful or even essen- 
tial in the analytic study of the behaviour of a mechanism, 
whether animate or inanimate. But its usefulness in the labora- 
tory should not obscure the fact that the free-living animal is not 
subject to these constraints. 

Sometimes systems which seem at first sight to be one-way 
prove on closer examination to have feedback. Walking on a 
smooth pavement, for instance, seems to involve so little reference 
to the structures outside the body that the nervous system might 
seem to be producing its actions without reference to their effects. 
Tabes dorsalis, however, prevents incoming sensory impulses from 
reaching the brain while leaving the outgoing motor impulses un- 
affected. If walking were due simply to the outgoing motor 
impulses, the disease would cause no disturbance to walking. In 
fact, it upsets the action severely, and demonstrates that the 
incoming sensory impulses are really playing an essential, though 
hidden, part in the normal action. 

Sometimes the feedback can be demonstrated only with diffi- 
culty. Thus, Lloyd Morgan raised some ducklings in an incubator. 

4 The ducklings thoroughly enjoyed a dip. Each morning, 
at nine o'clock, a large black tray was placed in their pen, 
and on it a flat tin containing water. To this they eagerly 
ran, drinking and washing in it. On the sixth morning the 
tray and tin were given them in the usual way, but without any 
water. They ran to it, scooped at the bottom and made all 
the motions of the beak as if drinking. They squatted in it, 
dipping their heads, and waggling their tails as usual. For 
some ten minutes they continued to wash in non-existent 
water . . . ' 

Their behaviour might suggest that the stimuli of tray and tin 
were compelling the production of certain activities and that the 
results of these activities were having no back-effect. But further 
experiment showed that some effect was occurring : 

' The next day the experiment was repeated with the dry tin. 
Again they ran to it, shovelling along the bottom with their 
beaks, and squatting down in it. But they soon gave up. 
On the third morning they waddled up to the dry tin, and 
departed.' 

38 



THE ANIMAL AS MACHINE 3/12 

Their behaviour at first suggested that there was no feedback. 
But on the third day their change of behaviour showed that, in 
fact, the change in the bath had had some effect on them. 

The importance of feedback lies in the fact that systems which 
possess it have certain properties (S. 4/14) which cannot be shown 
by systems lacking it. Systems with feedback cannot adequately 
be treated as if they were of one-way action, for the feedback 
introduces properties which can be explained only by reference 
to the properties of the particular feedback used. (On the other 
hand a one-way system can, without error, be treated as if it 
contained feedback : we assume that one of the two actions is 
present but at zero degree (S. 2/3). In other words, systems 
without feedback are a sub-class of the class of systems with 
feedback.) 

3/12. As the organism and its environment are to be treated 
as a single system, the dividing line between 4 organism ' and 
4 environment ' becomes partly conceptual, and to that extent 
arbitrary. Anatomically and physically, of course, there is a 
unique and obvious distinction between the two parts of the sys- 
tem ; but if we view the system functionally, ignoring purely 
anatomical facts as irrelevant, the division of the system into 
4 organism ' and ' environment ' becomes vague. Thus, if a 
mechanic with an artificial arm is trying to repair an engine, 
then the arm may be regarded either as part of the organism that 
is struggling with the engine, or as part of the machinery with 
which the man is struggling. 

Once this flexibility of division is admitted, almost no bounds 
can be put to its application. The chisel in a sculptor's hand 
can be regarded either as a part of the complex biophysical mechan- 
ism that is shaping the marble, or it can be regarded as a part of 
the material which the nervous system is attempting to control. 
The bones in the sculptor's arm can be regarded either as part of 
the organism or as part of the ' environment ' of the nervous 
system. Variables within the body may justifiably be regarded 
as the ' environment ' of some other part. A child has to learn 
not only how to grasp a piece of bread, but how to chew without 
biting his own tongue ; functionally both bread and tongue are 
part of the environment of the cerebral cortex. But the environ- 
ments with which the cortex has to deal are sometimes even deeper 

39 



3/13 DESIGN FOR A BRAIN 

in the body than the tongue : the child has to learn how to play 
without exhausting itself utterly, and how to talk without getting 
out of breath. 

These remarks are not intended to confuse, but to show that 
later arguments (S. 17/4 and Chapter 18) are not unreasonable. 
There it is intended to treat one group of neurons in the cerebral 
cortex as the environment of another group. These divisions, 
though arbitrary, are justifiable because we shall always treat the 
system as a whole, dividing it into parts in this unusual way merely 
for verbal convenience in description. 

It should be noticed that from now on w the system ' means 
not the nervous system but the whole complex of the organism 
and its environment. Thus, if it should be shown that i the 
system ' has some property, it must not be assumed that this 
property is attributed to the nervous system : it belongs to the 
whole ; and detailed examination may be necessary to ascertain 
the contributions of the separate parts. 

3/13. In some cases the dynamic nature of the interaction 
between organism and environment can be made intuitively more 
obvious by using the device, common in physics, of regarding the 
animal as the centre of reference. In locomotion the animal 
would then be thought of as pulling the world past itself. Pro- 
vided we are concerned only with the relation between these two, 
and are not considering their relations to any third and inde- 
pendent body, the device will not lead to error. It was used 
in the i rider and bicycle ' example. 

By the use of animal-centred co-ordinates we can see that the 
animal has much more control over its environment than might at 
first seem possible. Thus when a dog puts its foot on a sharp and 
immovable stone, the latter does not seem particularly dynamic. 
Yet the dog can cause great changes in this environment — by 
moving its foot away. Again, while a frog cannot change air into 
water, a frog on the bank of a stream can, with one small jump, 
change its world from one ruled by the laws of mechanics to one 
ruled by the laws of hydrodynamics. 

Static systems (like the sharp stone) can always be treated as if 
dynamic (though not conversely), for we have only to use the 
device of S. 2/3 and treat the static variable as one which is 
undergoing change of zero degree. The dynamic view is therefore 

40 



THE ANIMAL AS MACHINE 3/14 

the more general. For this reason the environment will always 
be treated as wholly dynamic. 



Essential variables 

3/14. The biologist must view the brain, not as being the seat of 
the ' mind ', nor as something that ' thinks ', but, like every other 
organ in the body, as a specialised means to survival. We shall 
use the concept of ' survival ' repeatedly ; but before we can use 
it, we must, by S. 2/8, transform it to our standard form. What 
does it mean in terms of primary operations ? 

Physico-chemical systems may undergo the most extensive 
transformations without showing any change obviously equivalent 
to death, for matter and energy are indestructible. Yet the dis- 
tinction between a live horse and a dead one is obvious enough 
— they fetch quite different prices in the market. The distinc- 
tion must be capable of objective definition. 

It is suggested that the definition may be obtained in the fol- 
lowing way. That an animal should remain ' alive ' certain 
variables must remain within certain ' physiological ' limits. 
What these variables are, and what the limits, are fixed when 
we have named the species we are working with. In practice 
one does not experiment on animals in general, one experiments 
on one of a particular species. In each species the many physio- 
logical variables differ widely in their relevance to survival. 
Thus, if a man's hair is shortened from 4 inches to 1 inch, the 
change is trivial ; if his systolic blood-pressure drops from 120 mm. 
of mercury to 30, the change will quickly be fatal. 

Every species has a number of variables which are closely re- 
lated to survival and which are closely linked dynamically so 
that marked changes in any one leads sooner or later to marked 
changes in the others. Thus, if we find in a rat that the pulse- 
rate has dropped to zero, we can predict that the respiration 
rate will soon become zero, that the body temperature will soon 
fall to room temperature, and that the number of bacteria in the 
tissues will soon rise from almost zero to a very high number. 
These important and closely linked variables will be referred to as 
the essential variables of the animal. 

How are we to discover them, considering that we may not use 
borrowed knowledge but must find them by the method of 

41 



3/14 DESIGN FOR A BRAIN 

S. 2/8 ? There is no difficulty. Given a species, we observe 
what follows when members of the species are started from a 
variety of initial states. We shall find that large initial changes 
in some variables are followed in the system by merely transient 
deviations, while large initial changes in others are followed by 
deviations that become ever greater till the ' machine ' changes 
to something very different from what it was originally. The 
results of these primary operations will thus distinguish, quite 
objectively, the essential variables from the others. This dis- 
tinction may not be quite clear, for an animal's variables cannot 
be divided sharply into ' essential ' and ' not essential ' ; but 
exactness is not necessary here. All that is required is the ability 
to arrange the animal's variables in an approximate order of 
importance. Inexactness of the order is not serious, for nowhere 
will we use a particular order as a basis for particular deductions. 
We can now define ' survival ' objectively and in terms of a 
field : it occurs when a line of behaviour takes no essential variable 
outside given limits. 

References 

Bartlett, F. C. The measurement of human skill. British Medical Journal, 

1, 835 ; 14 June 1947. 
Jennings, H. S. Behavior of the lower organisms. New York, 1906. 
Morgan, C. Lloyd. Habit and instinct. London, 1896. 
Rein, H. Die physiologischen Grundlagen des Kreislaufkollapses. Archiv 

fur klinische Chirurgie, 189, 302 ; 1937. 
Starling, E. H. Principles of human physiology. London, 6th edition, 1933. 



42 



CHAPTER 4 

Stability 



4/1. The words ' stability ', ' steady state ', and ' equilibrium ' 
are used by a variety of authors with a variety of meanings, 
though there is always the same underlying theme. As we shall 
be much concerned with stability and its properties, an exact 
definition must be provided. 

The subject may be opened by a presentation of the three 
standard elementary examples. A cube resting with one face 
on a horizontal surface typifies ' stable ' equilibrium ; a sphere 
resting on a horizontal surface typifies ' neutral ' equilibrium ; 
and a cone balanced on its point typifies ' unstable ' equilibrium. 
With neutral and unstable equilibria we shall have little concern, 
but the concept of ' stable equilibrium ' will be used repeatedly. 

These three dynamic systems are restricted in their behaviour 
by the fact that each system contains a fixed quantity of energy, 
so that any subsequent movement must conform to this invari- 
ance. We, however, shall be considering systems which are 
abundantly supplied with free energy so that no such limitation 
is imposed. Here are two examples. 

The first is the Watt's governor. A steam-engine rotates a pair 
of weights which, as they are rotated faster, separate more widely 
by centrifugal action ; their separation controls mechanically 
the position of the throttle ; and the position of the throttle 
controls the flow of steam to the engine. The connections are 
arranged so that an increase in the speed of the engine causes a 
decrease in the flow of steam. The result is that if any transient 
disturbance slows or accelerates the engine, the governor brings 
the speed back to the usual value. By this return the system 
demonstrates its stability. 

The second example is the thermostat, of which many types 
exist. All, however, work on the same principle : a chilling of 
the bath causes a change which in its turn causes the heating to 
become more intense or more effective ; and vice versa. The 

43 



4/2 DESIGN FOR A BRAIN 

result is that if any transient disturbance cools or overheats the 
bath, the thermostat brings the temperature back to the usual 
value. By this return the system demonstrates its stability. 

4/2. An important feature of stability is that it does not refer 
to a material body or ' machine ' but only to some aspect of it. 
This statement may be proved most simply by an example showing 
that a single material body can be in two different equilibrial 
states at the same time. Consider a square card balanced exactly 
on one edge : to displacements at right angles to this edge the 
card is unstable ; to displacements exactly parallel to this edge 
it is, theoretically at least, stable. 

The example supports the thesis that we do not, in general, 
study physical bodies but only entities carefully abstracted from 
them. The concept of stability must therefore be defined in 
terms of the basic primary operations (S. 2/3). 

4/3. Consider next a corrugated surface, laid horizontally, with 
a ball rolling from a ridge down towards a trough. A photograph 
taken in the middle of its roll would look like Figure 4/3/1. We 
might think of the ball as being unstable because it has rolled away 
from the ridge, until we realise that we can also think of it as 
stable because it is rolling towards the trough. The duality shows 

we are approaching the concept in the 
wrong way. The situation can be made 
clearer if we remove the ball and consider 
only the surface. The top of the ridge, 
as it would affect the roll of a ball, is 
now recognised as a position of unstable 
equilibrium, and the bottom of the 
trough as a position of stability. We 
now see that, if friction is sufficiently 
marked for us to be able to neglect 
momentum, the system composed of 
the single variable 4 distance of the ball laterally ' is absolute and 
has a definite, permanent field, which is sketched in the Figure. 

From B the lines of behaviour diverge, but to A they converge. 
We conclude tentatively that the concept of ' stability ' belongs not 
to a material body but to a field. It is shown by a field if the lines 
of behaviour converge. (An exact definition is given in S. 4/8.) 

44 







A [ 


3 



Figure 4/3/1. 



STABILITY 



4/5 



4/4. This preliminary remark begins to justify the emphasis 
placed on absoluteness. Since stability is a feature of a field, 
and since only regular systems have unchanging fields (S. 19/16) 
it follows that to discuss stability in a system we must suppose 
that the system is regular : we cannot test the stability of a 
thermostat if some arbitrary interference continually upsets it. 
But regularity in the system is not sufficient. If a field had 
lines criss-crossing like those of Figure 2/15/2 we could not make 
any simple statement about them. Only when the lines have a 
smooth flow like those of Figures 4/5/1, 4/5/2 or 4/10/1 can a 
simple statement be made about them. And this property 
implies (S. 19/12) that the system must be absolute. 

4/5. To illustrate that the concept of stability belongs to a 
field, let us examine the fields of the previous examples. 

The cube resting on one face yields an absolute system which 
has two variables : 

(cc) the angle which the face makes with the horizontal, and 

(y) the rate at which this angle changes. 
(This system allows for the momentum of the cube.) If the cube 
does not bounce when the face meets the table, the field is similar 




O O o a o o o 

Figure 4/5/1 : Field of the two-variable system described in the text. 
Below is shown the cube as it would appear in elevation when its main 
face, shown by a heavier line, is tilted through the angle x. 

45 





Y v 



Figure 4/5/2. 



4/5 DESIGN FOR A BRAIN 

to that sketched in Figure 4/5/1. The stability of the cube 
when resting on a face corresponds in the field to the convergence 
of the lines of behaviour to the centre. 

The square card balanced on its edge can be represented approxi- 
mately by two variables which measure displacements at right 

angles (x) and parallel (y) to the lower 
edge. The field will resemble that 
sketched in Figure 4/5/2. Displace- 
ment from the origin to A is followed 
by a return of the representative point 
to 0, and this return corresponds to the 
stability. Displacement from to B is 
followed by a departure from the region 
under consideration, and this departure 
corresponds to the instability. The 
uncertainty of the movements near O 
corresponds to the uncertainty in the behaviour of the card when 
released from the vertical position. 

The Watt's governor has a more complicated field, but an 
approximation may be obtained without difficulty. The system 
may be specified to an approximation sufficient for our purpose 
by three variables : 

(x) the speed of the engine and 

governor (r.p.m.), 
(y) the distance between the 
weights, or the position 
of the throttle, and 
(z) the velocity of flow of the 
steam. 
(y represents either of two quan- 
tities because they are rigidly 
connected). If, now, a disturb- 
ance suddenly accelerates the 
engine, increasing x, the increase 
in x will increase y ; this increase 
in y will be followed by a decrease 
of z, and then by a decrease of x. 
As the changes occur not in 
jumps but continuously, the line 
of behaviour must resemble that 

46 




Figure 4/5/3 : One line of behav- 
iour in the field of the Watt's 
governor. For clarity, the resting 
state of the system has been used 
as origin. The system has been 
displaced to A and then released, 



STABILITY 4/8 

sketched in Figure 4/5/3. The other lines of the field could be 
added by considering what would happen after other disturbances 
(lines starting from points other than A). Although having dif- 
ferent initial states, all the lines would converge towards 0. 

4/6. In some of our examples, for instance that of the cube, the 
lines of behaviour terminate in a point at which all movement 
ceases. In other examples the movement does not wholly cease ; 
many a thermostat settles down, when close to its resting state, 
to a regular small oscillation. We shall be little interested in the 
details of what happens at the exact centre. 

4/7. More important is the underlying theme that in all cases 
the stable system is characterised by the fact that after a displace- 
ment we can assign some limit to the subsequent movement of the 
representative point, whereas in the unstable system such limita- 
tion is either impossible or depends on facts outside the subject of 
discussion. Thus, if a thermostat is set at 37° C. and displaced 
to 40°, we can predict that in the future it will not go outside 
specified limits, which might be in one apparatus 36° and 40°. 
On the other hand, if the thermostat has been assembled with a 
component reversed so that it is unstable (S. 4/12) and if it is 
displaced to 40°, then we can give no limits to its subsequent 
temperatures ; unless we introduce such new topics as the melting- 
point of its solder. 

4/8. These considerations bring us to the definition which will 
be used. Given an absolute system and a region within its field, 
a line of behaviour from a point within the region is stable if it 
never leaves the region. Within one absolute system a change of 
the region or of the line of behaviour may change the result of 
the criterion. 

Thus, in Figure 4/3/1 the stability around A can be decided 
thus : make a mark on each side oi" A so as to define the region ; 
then as the line of behaviour from any point within this region 
never leaves it, the line of behaviour is stable. On the other 
hand, no region can be found around B which gives a stable line 
of behaviour. Again, consider Figure 4/5/2 : a boundary line 
is first drawn to enclose A, and B, in order to define which 
part of the field is being discussed. The line of behaviour from 

47 



4/9 DESIGN FOR A BRAIN 

A is then found to be stable, and the line from B unstable. This 
example makes it obvious that the concept of ' stability ' belongs 
primarily to a line of behaviour, not to a whole field. In particular 
it should be noted that in all cases the definition gives a unique 
answer once the line, the region, and the initial state are given. 

The examples above have been selected to test the definition 
severely. Sometimes the fields are simpler. In the field of the 
cube, for instance, it is possible to draw many boundaries, each oval 
in shape, such that all lines within the boundary are stable. The 
field of the Watt's governor is also of this type. It will be noticed 
that before we can discuss stability in a particular case we must 
always define which region of the phase-space we are referring to. 

A field within a given region is ' stable ' if every line of behaviour 
in the region is stable. A system is ' stable ' if its field is stable. 



4/9. A resting state is one from which an absolute system does 
not move when released. Such states occur in Figure 4/3/1 at 
A and B, and in Figure 4/5/1 at the origin. 

Although the variables do not change value when at a resting 
state this invariance does not imply that the ' machine ' itself is 
inactive. Thus, a steady Watt's governor implies that the engine 
is working at a non-zero rate. And a living muscle, even if 
unchanging in tension, is continually active in metabolism. 
4 Resting ' applies to the variables, not necessarily to the ' machine ' 

that yields the variables. 

4/10. If a line of behaviour is 
re-entrant to itself, the system 
undergoes a recurrent cycle. If 
the cycle is wholly contained in 
a given region, and the lines of 
behaviour lead into the cycle, the 
cycle is stable. 

Such a cycle is commonly shown 
by thermostats which, after correct- 
ing any gross displacement, settle 
down to a steady oscillation. In 
such a case the field will show, 

not convergence to a point but convergence to a cycle, such as is 

shown exaggerated in Figure 4/10/1. 

48 




Figure 4/10/1. 



STABILITY 4/12 

4/11. This definition of stability conforms to the requirement 
of S. 2/8 ; for the observed behaviour of the system determines 
the field, and the field determines the stability. 



Feedback 

4/12. The description given in S. 4/1 of the working of the 
Watt's governor showed that it is arranged in a functional circuit : 
the chain of cause and effect is re-entrant. Thus if we represent 
4 A has a direct effect on B ' or ' A directly disturbs B ' by the 
symbol A — > B, then the construction of the Watt's governor may 
be represented by the diagram : 



Speed of 
engine 



Distance 
between 
weights 



Velocity 

of flow 

of steam 



(The number of variables named here is partly optional.) 



Lest the diagram should seem based on some metaphysical 
knowledge of causes and effects, its derivation from the actual 
machine, using only primary operations, will be described. 

Suppose the relation between ' speed of engine ' and 4 distance 
between weights ' is first investigated. The experimenter would 
fix the variable ' velocity of flow of steam '. Then he would try 
various speeds of the engine, and would observe how these changes 
affected the behaviour of ' distance between the weights '. He 
would find that changes in the speed of the engine were regularly 
followed by changes in the distance between the weights. He 
need know nothing of the nature of the ultimate physical linkages, 
but he would observe the fact. Then, still keeping i velocity of 
flow of steam ' constant, he would try various distances between 
the weights, and would observe the effect of such changes on the 
speed of the engine ; he would find them to be without effect. 

49 



4/12 DESIGN FOR A BRAIN 

He would thus have established that there is an arrow from left 
to right but not from right to left in 



Speed of 
engine 



Distance 
between 
weights 



This procedure could then be applied to the two variables 
' distance between weights ' and ' velocity of flow of steam ', 
while the other variable 4 speed of engine ' was kept constant. 
And finally the relations between the third pair could be established. 

The method is clearly general. To find the immediate effects 
in a system with variables A, B, C, D . . . take one pair, A and 
B say ; hold all other variables C, D . . . constant ; note B's 
behaviour when A starts, or is held, at A x ; and also its behaviour 
when A starts, or is held, at A 2 . If these behaviours of B are the 
same, then there is no immediate effect from A to B. But if the 
B's behaviours are unequal, and regularly depend on what value 
A starts from, or is held at, then there is an immediate effect, 
which we symbolise by A — > B. 

By interchanging A and B in the process we can test for 
B — > A . And by using other pairs in turn we can determine 
all the immediate effects. The process is clearly defined, and 
consists purely of primary operations. It therefore uses no 
borrowed knowledge. We shall frequently use this diagram of 
immediate effects. 

If A has an immediate effect on B, and B has an immediate 
effect on A, the relation will be represented by A ^± B. If A 
affects B, and B also affects C, but A does not affect C directly, the 
relation will be shown by A — > B — > C. If there is a sequence of 
arrows joined head to tail and we are not interested in the inter- 
mediate steps, the sequence may often be contracted without 
ambiguity to A — > C. The diagram will be used only for illustration 
and not for rigorous proofs, so further precision is not required. (It 
should be carefully distinguished from the diagram of ' ultimate ' 
effects, but this is not required yet and will be described in S. 14/6. 
At the moment we regard the concept of one variable ' having an 
effect ' on another as well understood. But the concept will 
be examined more closely, and given more precision, in S. 14/3.) 

A gas thermostat also shows a functional circuit or feedback ; 

50 



STABILITY 4/12 

for if it is controlled by a capsule which by its swelling moves a 
lever which controls the flow of gas to the heating flame, the 
diagram of immediate effects would be : 



Temperature 




Diameter 


of capsule 




of capsule 


t 


k 




> ' 


Size of 
gas flame 




Position 
of lever 


> 


k 




y 


Velocity 




Position 


of ga 


s flow 




of gas tap 



The reader should verify that each arrow represents a physical 
action which can be demonstrated if all variables other than the 
pair are kept constant. 

Another example is provided by ' reaction ' in a radio receiver. 
We can represent the action by two variables linked in two ways : 



Amplitude of 
oscillation of the 
anode-potential 




Amplitude of 

oscillation of the 

grid-potential 







The lower arrow represents the grid-potential's effect within the 
valve on the anode-current. The upper arrow represents some 
arrangement of the circuit by which fluctuation in the anode- 
potential affects the grid-potential. The effect represented by 
the lower arrow is determined by the valve- designer, that of the 
upper by the circuit-designer. 

Such systems whose variables affect one another in a circuit 
possess what the radio-engineer calls ' feedback ' ; they are also 
sometimes described as ' servo-mechanisms '. They are at least 
as old as the Watt's governor and may be older. But only during 
the last decade has it been realised that the possession of feedback 
gives a machine potentialities that are not available to a machine 
lacking it. The development occurred mainly during the last 
war, stimulated by the demand for automatic methods of control 

51 



4/13 DESIGN FOR A BRAIN 

of searchlight, anti-aircraft guns, rockets, and torpedoes, and 
facilitated by the great advances that had occurred in electronics. 
As a result, a host of new machines appeared which acted with 
powers of self-adjustment and correction never before achieved. 
Some of their main properties will be described in S. 4/14. 

The nature, degree, and polarity of the feedback has a decisive 
effect on the stability or instability of the system. In the Watt's 
governor or in the thermostat, for instance, the connection of a 
part in reversed position, reversing the polarity of action of one 
component on the next, may, and probably will, turn the system 
from stable to unstable. In the reaction circuit of the radio set, 
the stability or instability is determined by the quantitative rela- 
tion between the two effects. 

Instability in such systems is shown by the development of a 
' runaway '. The least disturbance is magnified by its passage 
round the circuit so that it is incessantly built up into a larger 




4^=^3 3^ 4 

Figure 4/12/1. 

and larger deviation from the resting state. The phenomenon 
is identical with that referred to as a ' vicious circle '. 

The examples shown have only a simple circuit. But more 
complex systems may have many interlacing circuits. If, for 
instance, as in S. 8/8, four variables all act on each other, the 
diagram of immediate effects would be that shown in Figure 
4/12/1 (A). It is easy to verify that such a system contains 
twenty interlaced circuits, two of which are shown at B and C. 

The further development of the theory of systems with feed- 
back cannot be made without mathematics. But here it is 
sufficient to note two facts : a system which possesses feedback 
is usually actively stable or actively unstable ; and whether it is 
stable or unstable depends on the quantitative details of the 
particular arrangement. 

4/13. It will be noticed that stability, as denned, in no way 
implies fixity or rigidity. It is true that stable systems may have 
a resting state at which they will show no change ; but the lack 

52 



STABILITY 



4/14 



of change is deceptive if it suggests rigidity : they have only to be 
disturbed to show that they are capable of extensive and active 
movements. They are restricted only in that they do not show 
the unlimited divergencies of instability. 

Goal-seeking 

4/14. Every stable system has the property that if displaced 
from a resting state and released, the subsequent movement is 
so matched to the initial displacement that the system is brought 
back to the resting state. A variety of disturbances will therefore 
evoke a variety of matched reactions. Reference to a simple field 
such as that of Figure 4/5/1 will establish the point. 

This pairing of the line of return to the initial displacement 
has sometimes been regarded as ' intelligent ' and peculiar to living 
things. But a simple refutation is given by the ordinary pen- 
dulum : if we displace it to the right, it develops a force which 
tends to move it to the left ; and if we displace it to the left, it 
develops a force which tends to move it to the right. Noticing 




— Time 



Figure 4/14/1 : Tracing of the temperature (solid line), of a thermostatically 
controlled bath, and of the control setting (broken line). 

that the pendulum reacted with forces which though varied in 
direction always pointed towards the centre, the mediaeval scien- 
tist would have said ' the pendulum seeks the centre '. By this 
phrase he would have recognised that the behaviour of a stable 
system may be described as ' goal-seeking '. Without introducing 
any metaphysical implications we may recognise that this type of 
behaviour does occur in the stable dynamic systems. Thus 
Figure 4/14/1 shows how, as the control setting of a thermostat 
was altered, the temperature of the apparatus always followed it, 
the set temperature being treated as if it were a goal. 

Such a movement occurs here in only one dimension (tempera- 

53 E 



4/15 DESIGN FOR A BRAIN 

ture), but other goal-seeking devices may use more. The radar- 
controlled searchlight, for example, uses the reflected impulses 
to alter its direction of aim so as to minimise the angle between 
its direction of aim and the bearing of the source of the reflected 
impulses. So if the aircraft swerves, the searchlight will follow 
it actively, just as the temperature followed the setting. 

The examples show the common feature that each is ' error- 
controlled ' : each is partly controlled by the deviation of the 
system's state from the resting state (which, in these examples, 
can be moved by an outside operation). The thermostat is 
affected by the difference between the actual and the set tem- 
peratures. The searchlight is affected by the difference between the 
two directions. So it will be seen that machines with feedback are 
not subject to the oft-repeated dictum that machines must act 
blindly and cannot correct their errors. Such a statement is true 
of machines without feedback, but not of machines in general. 

Once it is appreciated that feedback can be used to correct any 
deviation we like, it is easy to understand that there is no limit 
to the complexity of goal- seeking behaviour which may occur in 
machines quite devoid of any ' vital ' or ' intelligent ' factor. 
Thus, an automatic anti-aircraft gun may be controlled by the 
radar-pulses reflected back both from the target aeroplane and 
from its own bursting shells, in such a way that it tends to mini- 
mise the distance between shell-burst and plane. Such a system, 
wholly automatic, cannot be distinguished by its behaviour from 
a humanly operated gun : both will fire at the target, following 
it through all manoeuvres, continually using the errors to improve 
the next shot. It will be seen, therefore, that a system with feed- 
back may be both wholly automatic and yet actively and complexly 
goal-seeking. There is no incompatibility. 

4/15. An important feature of a system's stability (or instability) 
is that it is a property of the whole system and can be assigned 
to no part of it. The statement may be illustrated by a con- 
sideration of the third diagram of S. 4/12 as it is related to the 
practical construction of the thermostat. In order to ensure the 
stability of the final assembly, the designer must consider : 
(1) The effect of the temperature on the diameter of the cap- 
sule, i.e. whether a rise in temperature makes the capsule 
expand or shrink. 

54 



STABILITY 4/16 

(2) Which way an expansion of the capsule moves the lever. 

(3) Which way a movement of the lever moves the gas-tap. 

(4) Whether a given movement of the gas-tap makes the 

velocity of gas-flow increase or decrease. 

(5) Whether an increase of gas-flow makes the size of the gas- 

flame increase or decrease. 

(6) How an increase in size of the gas-flame will affect the tem- 

perature of the capsule. 

Some of the answers are obvious, but they must none the less 
be included. When the six answers are known, the designer can 
ensure stability only by arranging the components (chiefly by 
manipulating (2), (3) and (5) ) so that as a whole they form an 
appropriate combination. Thus five of the effects may be decided, 
yet the stability will still depend on how the sixth is related to 
them. The stability belongs only to the combination ; it cannot 
be related to the parts considered separately. 

In order to emphasise that the stability of a system is inde- 
pendent of any conditions which may hold over the parts which 
compose the whole, some further examples will be given. (Proofs 
of the statements will be found in S. 21/5-7.) 

(a) Two systems may be joined so that they act and interact 
on one another to form a single system : to know that the two 
systems when separate were both stable is to know nothing about 
the stability of the system formed by their junction : it may be 
stable or unstable. 

(b) Two systems, both unstable, may join to form a whole which 
is stable. 

(c) Two systems may form a stable whole if joined in one way, 
and may form an unstable whole if joined in another way. 

(d) In a stable system the effect of fixing a variable may be to 
render the remainder unstable. 

Such examples could be multiplied almost indefinitely. They 
illustrate the rule that the stability (or instability) of a dynamic 
system depends on the parts and their interrelations as a whole. 

4/16. The fact that the stability of a system is a property of the 
system as a whole is related to the fact that the presence of stability 
(as contrasted with instability) always implies some co-ordination 
of the actions between the parts. In the thermostat the necessity 
for co-ordination is clear, for if the components were assembled 

55 



4/16 DESIGN FOR A BRAIN 

at random there would be only an even chance that the assembly 
would be stable. But as the system and the feedbacks become 
more complex, so does the achievement of stability become more 
difficult and the likelihood of instability greater. Radio engineers 
know only too well how readily complex systems with feedback 
become unstable, and how difficult is the discovery of just that 
combination of parts and linkages which will give stability. 

The subject is discussed more fully in S. 20/12 : here it is 
sufficient to note that as the number of variables increases so 
usually do the effects of variable on variable have to be co- 
ordinated with more and more care if stability is to be achieved. 



Reference 
Wiener, Norbert. Cybernetics. New York, 1948. 



56 



CHAPTER 5 

Adaptation as Stability 



5/1. The concept of ' adaptation ' has so far been used without 
definition ; this vagueness must be corrected. Not only must 
the definition be precise, but it must be given in terms that 
conform to the demand of S. 2/8. 

5/2. The suggestion that an animal's behaviour is ' adaptive ' 
if the animal ' responds correctly to a stimulus ' may be rejected 
at once. First, it presupposes an action by an experimenter and 
therefore cannot be applied when the free-living organism and 
its environment affect each other reciprocally. Secondly, the 
definition provides no meaning for 4 correctly ' unless it means 
4 conforming to what the experimenter thinks the animal ought 
to do '. Such a definition is useless. 



Homeostasis 

5/3. I propose the definition that a form of behaviour is adaptive 
if it maintains the essential variables (S. 3/14) within physiological 
limits. The full justification of such a definition would involve 
its comparison with all the known facts — an impossibly large 
task. Nevertheless it is fundamental in this subject and I must 
discuss it sufficiently to show how fundamental it is and how 
wide is its applicability. 

First I shall outline the facts underlying Cannon's concept of 
4 homeostasis '. They are not directly relevant to the problem 
of learning, for the mechanisms are inborn ; but the mechanisms 
are so clear and well known that they provide an ideal basic 
illustration. They show that : 

(1) Each mechanism is ' adapted ' to its end. 

(2) Its end is the maintenance of the values of some essential 

variables within physiological limits. 
57 



5/4 DESIGN FOR A BRAIN 

(3) Almost all the behaviour of an animal's vegetative system 
is due to such mechanisms. 

5/4. As first example may be quoted the mechanisms which 
tend to maintain within limits the concentration of glucose in 
the blood. The concentration should not fall below about 06 
per cent or the tissues will be starved of their chief source of 
energy; and the concentration should not rise above about 
0-18 per cent or other undesirable effects will occur. If the 
blood-glucose falls below about 0-07 per cent the adrenal glands 
secrete adrenaline, which makes the liver turn its stores of glycogen 
into glucose ; this passes into the blood and the fall is opposed. 
In addition, a falling blood-glucose stimulates the appetite so that 
food is taken, and this, after digestion, provides glucose. On 
the other hand, if it rises excessively, the secretion of insulin by 
the pancreas is increased, causing the liver to remove glucose 
from the blood. The muscles and skin also remove it ; and the 
kidneys help by excreting glucose into the urine if the concentra- 
tion in the blood exceeds 0-18 per cent. Here then are five 
activities all of which have the same final effect. Each one 
acts so as to restrict the fluctuations which might otherwise occur. 
Each may justly be described as ' adaptive ', for it acts to preserve 
the animal's life. 

The temperature of the interior of the warm-blooded animal's 
body may be disturbed by exertion, or illness, or by exposure 
to the weather. If the body temperature becomes raised, the 
skin flushes and more heat passes from the body to the sur- 
rounding air ; sweating commences, and the evaporation of the 
water removes heat from the body : and the metabolism of 
the body is slowed, so that less heat is generated within it. If the 
body is chilled, these changes are reversed. Shivering may start, 
and the extra muscular activity provides heat which warms the 
body. Adrenaline is secreted, raising the muscular tone and the 
metabolic rate, which again supplies increased heat to the body. 
The hairs or feathers are moved by small muscles in the skin so 
that they stand more erect, enclosing more air in the interstices 
and thus conserving the body's heat. In extreme cold the human 
being, when almost unconscious, reflexly takes a posture of 
extreme flexion with the arms pressed firmly against the chest 
and the legs fully drawn up against the abdomen. The posture 

58 



ADAPTATION AS STABILITY 5/4 

is clearly one which exposes to the air a minimum of surface. 
In all these ways, the body acts so as to maintain its temperature 
within limits. 

The water content of the blood is disturbed by the intake of 
water at drinking and eating, by the output during excretion 
and secretion, and by sweating. When the water content is 
lowered, sweating, salivation, and the excretion of urine are all 
diminished ; thirst is increased, leading to an increased intake, 
and the tissues of the body pass some of their water into the 
blood-stream. When the water content is excessive, all these 
activities are reversed. By these means the body tends to 
maintain the water-content of the blood within limits. 

The pressure of the blood in the aorta may be disturbed by 
haemorrhage or by exertion. When the pressure falls, centres 
in the brain and spinal cord make the heart beat faster, increasing 
the quantity of blood forced into the aorta ; they make the small 
arteries contract, impeding the flow of blood out of it. If the 
pressure is too high, these actions are reversed. By these and 
other mechanisms the blood pressure in the aorta is maintained 
within limits. 

The amount of carbon dioxide in the blood is important in 
its effect on the blood's alkalinity. If the amount rises, the rate 
and depth of respiration are increased, and carbon dioxide is 
exhaled at an increased rate. If the amount falls, the reaction 
is reversed. By this means the alkalinity of the blood is kept 
within limits. 

The retina works best at a certain intensity of illumination. 
In bright light the nervous system contracts the pupil, and in 
dim relaxes it. Thus the amount of light entering the eye is 
maintained within limits. 

If the eye is persistently exposed to bright light, as happens 
when one goes to the tropics, the pigment-cells in the retina 
grow forward day by day until they absorb a large portion 
of the incident light before it reaches the sensitive cells. In 
this way the illumination on the sensitive cells is kept within 
limits. 

If exposed to sunshine, the pigment-bearing cells in the skin 
increase in number, extent, and pigment-content. By this change 
the degree of illumination of the deeper layers of the skin is kept 
within limits. 

59 



5/5 DESIGN FOR A BRAIN 

When dry food is chewed, a copious supply of saliva is poured 
into the mouth. Saliva lubricates the food and converts it from 
a harsh and abrasive texture to one which can be chewed without 
injury. The secretion therefore keeps the frictional stresses 
below the destructive level. 

The volume of the circulating blood may be disturbed by 
haemorrhage. Immediately after a severe haemorrhage a number 
of changes occur : the capillaries in limbs and muscles undergo 
constriction, driving the blood from these vessels to the more 
essential internal organs ; thirst becomes extreme, impelling the 
subject to obtain extra supplies of fluid ; fluid from the tissues 
passes into the blood-stream and augments its volume ; and 
clotting at the wound helps to stem the haemorrhage. A haemor- 
rhage has a second effect in that, by reducing the number of 
red corpuscles, it reduces the amount of oxygen which can be 
carried to the tissues ; the reduction, however, itself stimulates 
the bone-marrow to an increased production of red corpuscles. 
All these actions tend to keep the variables ' volume of circulat- 
ing blood ' and ' oxygen supplied to the tissues ' within normal 
limits. 

Every fast-moving animal is liable to injury by collision with 
hard objects. Animals, however, are provided with reflexes that 
tend to minimise the chance of collision and of mechanical injury. 
A mechanical stress causes injury — laceration, dislocation, or 
fracture — only if the stress exceeds some definite value, depend- 
ing on the stressed tissue — skin, ligament, or bone. So these 
reflexes act to keep the mechanical stresses within physiological 
limits. 

Many more examples could be given, but all can be included 
within the same formula. Some external disturbance tends to 
drive an essential variable outside its normal limits ; but the 
commencing change itself activates a mechanism that opposes 
the external disturbance. By this mechanism the essential 
variable is maintained within limits much narrower than would 
occur if the external disturbance were unopposed. The nar- 
rowing is the objective form of the mechanism's adaptation. 

5/5. The mechanisms of the previous section act mostly within 
the body, but it should be noted that some of them have acted 
partly through the environment. Thus, if the body-temperature 

60 



ADAPTATION AS STABILITY 5/6 

is raised, the nervous system lessens the generation of heat within 
the body and the body-temperature falls, but only because the 
body is continuously losing heat to its surroundings. Flushing 
of the skin cools the body only if the surrounding air is cool ; 
and sweating lowers the body-temperature only if the surround- 
ing air is unsaturated. Increasing respiration lowers the carbon 
dioxide content of the blood, but only if the atmosphere contains 
less than 5 per cent. In each case the chain of cause and effect 
passes partly through the environment. The mechanisms that 
work wholly within the body and those that make extensive use 
of the environment are thus only the extremes of a continuous 
series. Thus, a thirsty animal seeks water : if it is a fish it does 
no more than swallow, while if it is an antelope in the veldt it 
has to go through an elaborate process of search, of travel, and 
of finding a suitable way down to the river or pond. The homeo- 
static mechanisms thus extend from those that work wholly 
within the animal to those that involve its widest-ranging activi- 
ties ; the principles are uniform throughout. 

5/6. Just the same criteria for ' adaptation ' may be used in 
judging the behaviour of the free-living animal in its learned 
reactions. Take the type-problem of the kitten and the fire. 
When the kitten first approaches an open fire, it may paw at the 
fire as if at a mouse, or it may crouch down and start to ' stalk ' 
the fire, or it may attempt to sniff at the fire, or it may walk un- 
concernedly on to it. Every one of these actions is liable to lead 
to the animal's being burned. Equally the kitten, if it is cold, 
may sit far from the fire and thus stay cold. The kitten's 
behaviour cannot be called adapted, for the temperature of its 
skin is not kept within normal limits. The animal, in other 
words, is not acting homeostatically for skin temperature. 
Contrast this behaviour with that of the experienced cat : on 
a cold day it approaches the fire to a distance adjusted so that 
the skin temperature is neither too hot nor too cold. If the fire 
burns fiercer, the cat will move away until the skin is again warmed 
to a moderate degree. If the fire burns low the cat will move 
nearer. If a red-hot coal drops from the fire the cat takes such 
action as will keep the skin temperature within normal limits. 
Without making any enquiry at this stage into what has hap- 
pened to the kitten's brain, we can at least say that whereas 

61 



5/6 DESIGN FOR A BRAIN 

at first the kitten's behaviour was not homeostatic for skin 
temperature, it has now become so. Such behaviour is ' adapted ' : 
it preserves the life of the animal by keeping the essential variables 
within limits. 

The same thesis can be applied to a great deal, if not all, of 
the normal human adult's behaviour. In order to demonstrate 
the wide application of this thesis, and in order to show that 
even Man's civilised life is not exceptional, some of the surround- 
ings which he has provided for himself will be examined for their 
known physical and physiological effects. It will be shown that 
each item acts so as to narrow the range of variation of his 
essential variables. 

The first requirement of a civilised man is a house ; and its 
first effect is to keep the air in which he lives at a more equable 
temperature. The roof keeps his skin at a more constant dryness. 
The windows, if open in summer and closed in winter, assist in 
the maintenance of an even temperature, and so do fires and 
stoves. The glass in the windows keeps the illumination of the 
rooms nearer the optimum, and artificial lighting has the same 
effect. The chimneys keep the amount of irritating smoke in 
the rooms near the optimum, which is zero. 

Many of the other conveniences of civilisation could, with little 
difficulty, be shown to be similarly variation-limiting. An attempt 
to demonstrate them all would be interminable. But to confirm 
the argument we will examine a motor-car, part by part, in 
order to show its homeostatic relation to man. 

Travel in a vehicle, as contrasted with travel on foot, keeps 
several essential variables within narrower limits. The fatigue 
induced by walking for a long distance implies that some vari- 
ables, as yet not clearly known, have exceeded limits not trans- 
gressed when the subject is carried in a vehicle. The reserves 
of food in the body will be less depleted, the skin on the soles 
of the feet will be less chafed, the muscles will have endured 
less strain, in winter the body will have been less chilled, and 
in summer it will have been less heated, than would have hap- 
pened had the subject travelled on foot. 

When examined in more detail, many ways are found in which 
it serves us by maintaining our essential variables within narrower 
limits. The roof maintains our skin at a constant dryness. The 
windows protect us from a cold wind, and if open in summer, 

62 



ADAPTATION AS STABILITY 5/7 

help to cool us. The carpet on the floor acts similarly in winter, 
helping to prevent the temperature of the feet from falling below 
its optimal value. The jolts of the road cause, on the skin and 
bone of the human frame, stresses which are much lessened by 
the presence of springs. Similar in action are the shock-absorbers 
and tyres. A collision would cause an extreme deceleration 
which leads to very high values for the stress on the skin and 
bone of the passengers. By the brakes these very high values 
may be avoided, and in this way the brakes keep the variables 
4 stress on bone ' within narrower limits. Good headlights keep 
the luminosity of the road within limits narrower than would 
occur in their absence. 

The thesis that ' adaptation ' means the maintenance of essential 
variables within physiological limits is thus seen to hold not 
only over the simpler activities of primitive animals but over 
the more complex activities of the ' higher ' organisms. 

5/7. Before proceeding further, it must be noted that the word 
' adaptation ' is commonly used in two senses which refer to 
different processes. 

The distinction may best be illustrated by the inborn homeo- 
static mechanisms : the reaction to cold by shivering, for instance. 
Such a mechanism may undergo two types of ' adaptation '. 
The first occurred long ago and was the change from a species 
too primitive to show such a reaction to a species which, by 
natural selection, had developed the reaction as a characteristic 
inborn feature. The second type of ' adaptation ' occurs when 
a member of the species, born with the mechanism, is subjected 
to cold and changes from not-shivering to shivering. The first 
change involved the development of the mechanism itself ; the 
second change occurs when the mechanism is stimulated into 
showing its properties. 

In the learning process, the first stage occurs when the animal 
1 learns ' : when it changes from an animal not having an adapted 
mechanism to one which has such a mechanism. The second 
stage occurs when the developed mechanism changes from in- 
activity to activity. In this chapter we are concerned with the 
characteristics of the developed mechanism. The processes 
which led to its development are discussed in Chapter 8. 

63 



5/8 DESIGN FOR A BRAIN 

5/8. We can now recognise that 4 adaptive ' behaviour is equi- 
valent to the behaviour of a stable system, the region of the 
stability being the region of the phase-space in which all the 
essential variables lie within their normal limits. 

The view is not new (though it can now be stated with more 
precision) : 

1 Every phase of activity in a living being must be not 
only a necessary sequence of some antecedent change in its 
environment, but must be so adapted to this change as to 
tend to its neutralisation, and so to the survival of the 
organism. ... It must also apply to all the relations of 
living beings. It must therefore be the guiding principle, 
not only in physiology . . . but also in the other branches 
of biology which treat of the relations of the living animal 
to its environment and of the factors determining its survival 
in the struggle for existence.' 

(Starling.) 

1 In an open system, such as our bodies represent, com- 
pounded of unstable material and subjected continuously to 
disturbing conditions, constancy is in itself evidence that 
agencies are acting or ready to act, to maintain this con- 
stancy.' 

(Cannon.) 

4 Every material system can exist as an entity only so long 
as its internal forces, attraction, cohesion, etc., balance the 
external forces acting upon it. This is true for an ordinary 
stone just as much as for the most complex substances ; 
and its truth should be recognised also for the animal organism. 
Being a definite circumscribed material system, it can only 
continue to exist so long as it is in continuous equilibrium 
with the forces external to it : so soon as this equilibrium 
is seriously disturbed the organism will cease to exist as the 
entity it was.' 

(Pavlov.) 

McDougall never used the concept of 4 stability ' explicitly, but 
when describing the type of behaviour which he considered to 
be most characteristic of the living organism, he wrote : 

c Take a billiard ball from the pocket and place it upon the 
table. It remains at rest, and would continue to remain so 
for an indefinitely long time, if no forces were applied to it. 
Push it in any direction, and its movement in that direction 
persists until its momentum is exhausted, or until it is 
deflected by the resistance of the cushion and follows a new 

64 



ADAPTATION AS STABILITY 5/11 

path mechanically determined. . . . Now contrast with this 
an instance of behaviour. Take a timid animal such as a 
guinea-pig from its hole or nest, and put it upon the grass 
plot. Instead of remaining at rest, it runs back to its hole ; 
push it in any other direction, and, as soon as you withdraw 
your hand, it turns back towards its hole ; place any obstacle 
in its way, and it seeks to circumvent or surmount it, rest- 
lessly persisting until it achieves its end or until its energy 
is exhausted.' 

He could hardly have chosen an example showing more clearly 
the features of stability. 



Survival 

5/9. The forces of the environment, and even the drift of time, 
tend to displace the essential variables by amounts to which we 
can assign no limit. For survival, the' essential variables must 
be kept within their physiological limits. In other words, the 
values of the essential variables must stay within some definite 
region in the system's phase-space. It follows therefore that 
unless the environment is wholly inactive, stability is necessary 
for survival. 

5/10. If an animal's behaviour always maintains its essential 
variables within their physiological limits, then the animal can 
die only of old age. Disease might disturb the essential variables, 
but the processes of repair and immunity would tend to restore 
them. But it is equally clear that the environment sometimes 
causes disturbances for which the body's stabilising powers are 
inadequate ; infections may prove too virulent, cold too extreme, 
a famine too severe, or the attack of an enemy too swift. 

The possession of a mechanism which stabilises the essential 
variables is therefore of advantage : against moderate disturb- 
ances it may be life-saving even if it eventually fails at some 
severe disturbance. It promotes, but does not guarantee, survival. 

5/11. Are there aspects of ' adaptation ' not included within the 
definition of ' stability ' ? Is ' survival ' to be the sole criterion 
of adaptation ? Is it to be maintained that the Roman soldier 
who killed Archimedes in Syracuse was better ' adapted ' in his 
behaviour than Archimedes ? 

65 



5/11 DESIGN FOR A BRAIN 

The question is not easily answered. It is similar to that of 
S. 3/4 where it was asked whether all the qualities of the living 
organism could be represented by number ; and the answer must 
be similar. It is assumed that we are dealing primarily with 
the simpler rather than with the more complex creatures, though 
the examples of S. 5/6 have shown that some at least of man's 
activities may be judged properly by this criterion. 

In order to survey rapidly the types of behaviour of the more 
primitive animals, we may examine the classification of Holmes, 
who intended his list to be exhaustive but constructed it with 
no reference to the concept of stability. The reader will be able 
to judge how far our formulation (S. 5/8) is consistent with his 
scheme, which is given in Table 5/11/1. 



Behaviour - 



r Non-adaptive 



Sustentative 



f Self- 
maintaining, 



Adaptive- 



Race- 
maintaining 



Useless tropistic reaction. 
Misdirected instinct. 
Abnormal sex behaviour. 
Pathological behaviour. 
Useless social activity. 
Superfluous random 
movements. 

Capture, devouring of 

food. 
Activities preparatory, as 

making snares, stalking. 
Collection of food, digging. 
Migration. 
Caring for food, storing, 

burying, hiding. 
Preparation of food. 

Against enemies — fight, 

flight. 
Against inanimate forces. 
Reactions to heat, gra vity, 

chemicals. 
^Against inanimate objects. 

Ameliorative Rest, sleep, play, basking. 

(with these we are not concerned, 



Protective 



S. 1/3). 
Table 5/11/1 : All forms of animal behaviour, classified by Holmes. 



For the primitive organism, and excluding behaviour related to 
racial survival, there seems to be little doubt that the w adaptive- 
ness ' of behaviour is properly measured by its tendency to 
promote the organism's survival. 

66 



ADAPTATION AS STABILITY 5/13 

The stable organism 

5/12. A most impressive characteristic of living organisms is 
their mobility, their tendency to change. McDougall expressed 
this characteristic well in the example of S. 5/8. Yet our 
formulation transfers the centre of interest to the resting 
state, to the fact that the essential variables of the adapted 
organism change less than they would if they were unadapted. 
Which is important : constancy or change ? 

The two aspects are not incompatible, for the constancy of some 
variables may involve the vigorous activity of others. A good 
thermostat reacts vigorously to a small change of temperature, 
and the vigorous activity of some of its variables keeps the 
others within narrow limits. The point of view taken here is 
that the constancy of the essential variables is fundamentally 
important, and that the activity of the other variables is impor- 
tant only in so far as it contributes to this end. 

5/13. So far the discussion has traced the relation between the 
concepts of ' adaptation ' and of 4 stability '. It will now be 
proposed that ' motor co-ordination ' also has an essential con- 
nection with stability. 

c Motor co-ordination ' is a concept well understood in physio- 
logy, where it refers to the ability of the organism to combine the 
activities of several muscles 
so that the resulting move- 
ment follows accurately its 
appropriate path. Con- 
trasted to it are the concepts 
of clumsiness, tremor, ataxia, 
athetosis. It is suggested Figure 5/13/1. 

that the presence or absence 

of co-ordination may be decided, in accordance with our methods, 
by observing whether the movement does, or does not, deviate 
outside given limits. 

The formulation seems to be adequate provided that we measure 
the limb's deviations from some line which is given arbitrarily, 
usually by a knowledge of the line followed by the normal limb. 
A first example is given by Figure 5/13/1, which shows the line 
traced by the point of an expert fencer's foil during a lunge. 

67 




5/14 



DESIGN FOR A BRAIN 



Any inco-ordination would be shown by a divergence from the 

intended line. 

A second example is given by the record of Figure 5/13/2. 
The subject, a patient with a tumour in 
the left cerebellum, was asked to follow 
the dotted lines with a pen. The left- 
and right-hand curves were drawn with 
the respective hands. The tracing shows 
clearly that the co-ordination is poorer 
in the left hand. What criterion reveals 
the fact? The essential distinction is 
that the deviations of the lines from the 
dots are larger on the left than on the 
right. 

The degree of motor co-ordination 
achieved may therefore be measured by 
the smallness of the deviations from 
some standard line. Later it will be sug- 
gested that there are mechanisms which 
act to maintain variables within narrow 

limits. If the identification of this section is accepted, such 

mechanisms could be regarded as appropriate for the co-ordination 

of motor activity. 




Figure 5/13/2 : Record 
of the attempts of a 
patient to follow the 
dotted lines with the left 
and right hands. (By 
the courtesy of Dr. W. T. 
Grant of Los Angeles.) 



5/14. So far we have noticed in stable systems only their pro- 
perty of keeping variables within limits. But such systems have 
other properties of which we shall notice two. They are also 
shown by animals, and are then sometimes considered to provide 
evidence that the organism has some power of ' intelligence ' not 
shared by non-living systems. In these two instances the assump- 
tion is unnecessary. 

The first property is shown by a stable system when the lines 
of behaviour do not return directly, by a straight line, to the 
resting state (e.g. Figure 4/5/3). When this occurs, variables 
may be observed to move away from their values in the resting 
state, only to return to them later. Thus, suppose in Figure 
5/14/1 that the field is stable and that at the resting state R 
x and y have the values X and Y. For clarity, only one line 
of behaviour is drawn. Let the system be displaced to A and 
its subsequent behaviour observed. At first, while the repre- 

68 



ADAPTATION AS STABILITY 5/15 

sentative point moves towards B, y hardly alters ; but x, which 

started at X', moves to X and goes past it to X" . Then x remains 

almost constant and y changes until the representative point 

reaches C. Then y stops changing, 

and x changes towards, and reaches, 

its resting value X. The system has 

now reached its resting state and no 

further changes occur. This account 

is just a transcription into words of 

what the field defines graphically. 

Now the shape and features of „ 

\ Figure 5/14/1. 

any field depend ultimately on the 

real physical and chemical construction of the ' machine ' from 
which the variables are abstracted. The fact that the line of 
behaviour does not run straight from A to R must be due to 
some feature in the 4 machine ' such that if the machine is to 
get from state A to state R, states B and C must be passed 
through of necessity. Thus, if the machine contained moving 
parts, their shapes might prohibit the direct route from A to R ; 
or if the system were chemical the prohibition might be thermo- 
dynamic. But in either case, if the observer watched the machine 
work, and thought it alive, he might say : ' How clever ! x 
couldn't get from A to R directly because this bar was in the 
way ; so x went to B, which made y carry x from B to C ; and 
once at C, x could get straight back to R. I believe x shows 
foresight.' 

Both points of view are reasonable. A stable system may 
be regarded both as blindly obeying the laws of its nature, and 
also as showing a rudimentary skill in getting back to its resting 
state in spite of obstacles. 

5/15. The second property is shown when an organism reacts to 
a variable with which it is not directly in contact. Suppose, 
for instance, that the diagram of immediate effects (S. 4/12) is 
that of Figure 5/15/1 ; the variables have been divided by the 
dotted line into ' animal ' on the right and 4 environment ' on 
the left, and the animal is not in direct contact with the variable 
marked X. The system is assumed to be stable, i.e. to have 
arrived at the ' adapted ' condition (S. 5/7). If disturbed, its 
changes will show co-ordination of part with part (S. 5/14), and 

69 f 



5/16 



DESIGN FOR A BRAIN 



this co-ordination will hold over the whole system (S. 4/15). It 
follows that the behaviour of the ' animal '-part will be co- 
ordinated with the behaviour of X although the ' animal ' has 
no immediate contact with it. 

In the higher organisms, and especially in man, the power to 
react correctly to something not immediately visible or tangible 




-< — | , 

-< — | . 

i 

: ►-• 

■ >- . 




Figure 5/15/1. 

has been called i imagination ', or i abstract thinking ', or several 
other names whose precise meaning need not be discussed at 
the moment. Here we should notice that the co-ordination of 
the behaviour of one part with that of another part not in direct 
contact with it is simply an elementary property of the stable 
system. 



5/16. At this stage it is convenient to re-state our problem in 
the new vocabulary. If, for brevity, we omit minor qualifications, 
we can state it thus : A determinate ' machine ' changes from 
a form that produces chaotic, unadapted behaviour to a form in 
which the parts are so co-ordinated that the whole is stable, 
acting to maintain certain variables within certain limits — how 
can this happen ? For example, what sort of a thermostat could, 
if assembled at random, rearrange its own parts to get itself 
stable for temperature ? 

It will be noticed that the new statement involves the concept 
of a machine changing its internal organisation. So far, nothing 
has been said of this important concept ; so it will be treated 
in the next two chapters. 

70 



ADAPTATION AS STABILITY 5/16 



References 

Ashby, W. Ross. Adaptiveness and equilibrium. Journal of Mental Science, 

86, 478 ; 1940. 
Idem. The behavioral properties of systems in equilibrium. American 

Journal of Psychology, 59, 682 ; 1946. 
Cannon, W. B. The wisdom of the body. London, 1932. 
Conference on Teleological Mechanisms. Annals of the New York 

Academy of Science, 50, 187 ; 1948. 
Grant, W. T. Graphic methods in the neurological examination : wavy 

tracings to record motor control. Bulletin of the Los Angeles Neurological 

Society, 12, 104 ; 1947. 
Holmes, S. J. A tentative classification of the forms of animal behavior. 

Journal of Comparative Psychology, 2, 173 ; 1922. 
McDougall, W. Psychology. New York, 1912. 
Pavlov, I. P. Conditioned reflexes. Oxford, 1927. 
Rosenbluetii, A., Wiener, N., and Bigelow, J. Behavior, purpose and 

teleology. Philosophy of Science, 10, 18 ; 1943. 
Sommerhoff, G. Analytical biology. Oxford, 1950. 



71 



CHAPTER 6 

Parameters 



6/1. So far, we have discussed the changes shown by the vari- 
ables of an absolute system, and have ignored the fact that all its 
changes occur on a background, or on a foundation, of constancies. 
Thus, a particular simple pendulum provides two variables which 
are known (S. 2/15) to be such that, if we are given a particular 
state of the system, we can predict correctly its ensuing be- 
haviour ; what has not been stated explicitly is that this is true 
only if the length of the string remains constant. The background, 
and these constancies, must now be considered. 

Every absolute system is formed by selecting some variables out 
of the totality of possible variables. ' Forming a system ' 
means dividing all possible variables into two classes : those 
within the system and those without. These two types of variable 
are in no way different in their intrinsic physical nature, but they 
stand in very different relations to the system. 

6/2. Given a system, a variable not included in it will be 
described as a parameter. The word variable will, from now on, 
be reserved for one within the system. 

In general, given a system, the parameters will differ in their 
closeness of relation to it. Some will have a direct relation to it : 
their change of value would affect the system to a major degree ; 
such is the parameter ' length of pendulum ' in its relation to the 
two-variable system of the previous section. Some are less 
closely related to it, their changes producing only a slight effect 
on it ; such is the parameter ' viscosity of the air ' in relation to 
the same system. And finally, for completeness, may be men- 
tioned the infinite number of parameters that are without detect- 
able effect on the system ; such are the brightness of the light 
shining on the pendulum, the events in an adjacent room, and the 
events in the distant nebulae. Those without detectable effect 

72 



PARAMETERS 



6/3 



may be ignored ; but the relationship of an effective parameter 
to a system must be clearly understood. 

Given a system, the effective parameters are usually innumer- 
able, so that a list is bounded only by the imagination of the 
writer. Thus, parameters whose change might affect the be- 
haviour of the same system of two variables are : 

(1) the length of the pendulum (hitherto assumed constant), 

(2) the lateral velocity of the air (hitherto assumed to be 

constant at zero), 

(3) the viscosity of the surrounding medium (hitherto assumed 

constant), 

(4) the position (co-ordinates) of the point of support, 

(5) the force of gravity, 

(6) the magnetic field in which it swings, 

(7) the elastic constant of the string of the pendulum, 

(8) its electrostatic charge, and the charges on bodies nearby ; 
but the list has no end. 



Parameter and field 

6/3. The effect on an absolute system of a change of parameter- 
value will now be shown. Table 6/3/1 shows the results of four 



Length 
(cm.) 


Line 


Vari- 
able 


Time 





005 


010 


015 


0-20 


0-25 | 0-30 


40 


1 


X 

y 



147 


7 
142 


14 
129 


20 
108 


25 

80 


28 
48 


29 
12 


2 


X 

y 


14 
129 


20 

108 


25 

80 


28 
48 


29 
12 


29 
- 24 


27 

-58 


60 


3 


X 

y 



147 


7 
144 


14 
135 


21 
121 


26 
101 


31 

78 


34 
51 


4 


X 

y 


21 
121 


26 
101 


31 

78 


34 
51 


36 
23 


36 
- 6 


35 
- 36 



Table 6/3/1. 
73 




6/4 DESIGN FOR A BRAIN 

primary operations applied to the two-variable system mentioned 

above, x is the angular deviation from the vertical, in degrees ; 

y is the angular velocity, in degrees per second ; the time is in 

seconds. 

The first two Lines show that the lines of behaviour following 

the state x = 14, y = 129 are equal, so the system, as far as 

it has been tested, is absolute. 
The line of behaviour is shown 
solid in Figure 6/3/1. In these 

Iqq. swings the length of the pendulum 

was 40 cm. This parameter was 
then changed to 60 cm. and two 
further lines of behaviour were 
observed. On these two, the lines 
of behaviour following the state 
x = 21, y = 121 are equal, so the 

Figure 6/3/1. s y stem is a S ain absolute - The 

line of behaviour is shown dotted 

in the same figure. But the change of parameter- value has caused 

the line of behaviour from x = 0, y = 147 to change. 

The relationship which the parameter bears to the two variables 

is therefore as follows : 

(1) So long as the parameter is constant, the system of x and 
y is absolute and has a definite field. 

(2) After the parameter changes from one constant value to 
another, the system of x and y becomes again absolute, and has a 
definite field, but this field is not the same as the previous one. 

The relation is general. A change in the value of an effective 
parameter changes the line of behaviour from each state. From 
this follows at once : a change in the value of an effective parameter 
changes the field. 

The converse proposition is also true. Suppose we form a 
system's field and find it to be absolute. If our control of its 
surroundings has not been complete, and we test it later and find 
it to be again absolute but to have a changed field, then we may 
deduce, by S. 22/5, that some parameter must, in the interval, 
have changed from one constant value to another constant value. 

6/4. The importance of distinguishing between change of a 
variable and change of a parameter, that is, between change of 

74 



PARAMETERS 



6/4 



300- 



200 



state and change of field, can hardly be over-estimated. In 
order to make the distinction clear I will give some examples. 

In a working clock, the single variable defined by the reading 
of the minute-hand on the face is absolute as a one-variable system ; 
for after some observations of its behaviour, we can predict the 
line of behaviour which will follow any given state. If now the 
regulator (the parameter) is moved to a new position, so that the 
clock runs at a different rate, and the system is re-examined, it 
will be found to be still absolute but to have a different field. 

If a healthy person drinks 100 g. of glucose dissolved in water, 
the amount of glucose in his blood usually rises and falls as A 
in Figure 6/4/1. The single variable c blood -glucose ' is not 
absolute, for a given state 
(e.g. 120 mg./lOO ml.) does 
not define the subsequent 
behaviour, for the blood- 
glucose may rise or fall. 
By adding a second vari- 
able, however, such as ' rate 
of change of blood-glucose ', 
which may be positive or 
negative, we obtain a two- 
variable system which is 
sufficiently absolute for 
illustration. The field of 
this two-variable system 
will resemble that of A in 
Figure 6/4/2. But if the subject is diabetic, the curve of the 
blood-glucose, even if it starts at the same initial value, rises 
much higher, as B in Figure 6/4/1. When the field of this 
behaviour is drawn (B, Fig. 6/4/2), it is seen to be not the same 
as that of the normal subject. The change of value of the 
parameter 4 degree of diabetes present ' has thus changed the 
field. 

Girden and Culler developed a conditioned reflex in a dog which 
was under the influence of curare (a paralysing drug). When 
later the animal was not under its influence, the conditioned reflex 
could not be elicited. But when the dog was again put under its 
influence, the conditioned reflex returned. We need not enquire 
closely into the absoluteness of the system, but we note that two 

75 




2 3 

Figure 6/4/1 : Changes in blood-glucose 
after the ingestion of lOOg. of glucose : 
(A) in the normal person, (B) in the 
diabetic. 



6/5 



DESIGN FOR A BRAIN 



characteristic lines of behaviour (two responses to the stimulus) 
existed, and that one line of behaviour was shown when the 



3\£ 

St- 200- 


A 


r s 


B 


blood 
100 ml 

6 
o 








° -100' ^^.y 

o 


1 




<b 









a. '00 200 100 200 300 

BLOOD GLUCOSE (mg. per 100 ml.) 

Figure 6/4/2 : Fields of the two lines of behaviour, A and B, 
from Figure 6/4/1. Cross-strokes mark each quarter-hour. 

parameter ' concentration of curare in the tissues ' had a high 
value, and the other when the parameter had a low value. 

6/5. The physicist, studying systems whose variables are all 
clearly marked and controllable, seldom confuses change of state 
with change of field. The psychologist, however, studies systems 
whose variables, even in the simplest systems, are so numerous 
that he cannot, in practice, make an exact list of them : his 
grasp of the situation must be intuitive rather than explicit. 
In his practical work he seldom fails to distinguish between the 
variables he is observing and the parameters he is controlling ; 
it is chiefly in his theoretical work, especially when he discusses 
cerebral mechanisms, that he is apt to allow the distinction to 
become blurred. To preserve the distinction between variable 
and parameter we must discuss, not the real ' machine ', with 
its infinite richness of variables, but a defined system. The 
advantage to be gained will become clearer as we proceed. 



Stimuli 

6/6. Many stimuli may be represented adequately as a change 
of parameter-value, so it is convenient here to relate the physio- 
logical and psychological concept of a ' stimulus ' to our methods. 
In all cases the diagram of immediate effects is 

(experimenter) — > stimulator — > animal — > recorders. 

76 



PARAMETERS 6/6 

In some cases the animal, at some resting state, is subjected to 
a sudden change in the value of the stimulator, and the second 
value is sustained throughout the observation. Thus, the pupil- 
lary reaction to light is demonstrated by first accustoming the 
eye to a low intensity of illumination, and then suddenly raising 
the illumination to a high level which is maintained while the 
reaction proceeds. In such cases the stimulator is parameter to 
the system ' animal and recorders ' ; and the physiologist's 
comparison of the previous control-behaviour with the behaviour 
after stimulation is equivalent, in our method, to a comparison of 
the two lines of behaviour that, starting from the same initial 
state, run in the two fields provided by the two values of the 
stimulator. 

Sometimes a parameter is changed sharply and is immediately 
returned to its initial value, as when the experimenter applies a 
single electric shock, a tap on a tendon, or a flash of light. The 
effect of the parameter-change is a brief change of field which, 
while it lasts, carries the representative point away from its 
original position. When the parameter is returned to its original 
value, the original field and resting state are restored, and the 
representative point returns to the resting state. Such a stimulus 
reveals a line of behaviour leading to the resting state. 

It will be necessary later to be more precise about what we mean 
by ' the ' stimulus. Consider, for instance, a dog developing a 
conditioned reflex to the ringing of an electric bell. What is the 
stimulus exactly ? Is it the closing of the contact switch ? The 
intermittent striking of the hammer on the bell ? The vibrations 
in the air ? The vibrations of the ear-drum, of the ossicles, of 
the basilar membrane ? The impulses in the acoustic nerve, in the 
temporal cortex ? If we are to be precise we must recognise that the 
experimenter controls directly only the contact switch, and that 
this acts as parameter to the complexly-acting system of electric 
bell, middle ear, and the rest. 

When the 4 stimulus ' becomes more complex we must generalise. 
One generalisation increases the number of parameters made to 
alter, as when a conditioned dog is subjected to combinations of a 
ticking metronome, a smell of camphor, a touch on the back, and 
a flashing light. Here we should notice that if the parameters 
are not all independent but change in groups, like the variables 
in S. 3/3, we can represent each undivided group by a single 

77 



6/7 DESIGN FOR A BRAIN 

parameter and thus avoid using unnecessarily large numbers 
of parameters. 

A more extensive generalisation is provided if we replace 
4 change of parameter ' by ' change of initial state '. It will be 
shown (S. 7/7 and 21/4) that if a variable, or parameter, stays 
constant over some period it may, within the period, be regarded 
indifferently as inside or outside the system — as variable or para- 
meter. If, therefore, a contact switch, once set, stays as the 
experimenter leaves it, we may, if we please, regard it as part of 
the system. Then what was a comparison between two lines of 
behaviour from two fields (of a set of variables a, b, c, say) under 
the change of a parameter p from p' to p", becomes a comparison 
between two lines of behaviour of the four-variable system from 
the initial states a, b, c, p' and a, b, c, p" . 4 Applying a stimulus ' 
is now equivalent to ' releasing from a different initial state ' ; 
and this will be used as its most general representation. 



Parameter and stability 

6/7. We now reach the main point of the chapter. Because 
a change of parameter-value changes the field, and because a 
system's stability depends on its field, a change of parameter- 
value will in general change a system's stability in some way. 

A simple example is given by a mixture of hydrogen, nitrogen, 
and ammonia, which combine or dissociate until the concentra- 
tions reach the resting state. If the mixture was originally derived 
from pure ammonia, the single variable 4 percentage dissociated ' 
forms a one-variable absolute system. Among its parameters 
are temperature and pressure. As is well known, changes in these 
parameters affect the position of the resting state. 

Such a system is simple and responds to the changes of the 
parameters with only a simple shift of resting state. No such 
limitation applies generally. Change of parameter-value may 
result in any change which can be produced by the substitution of 
one field for another : stable systems may become unstable, 
resting states may be moved, single resting states may become 
multiple, resting states may become cycles ; and so on. Figure 
21/5/1 provides an illustration. 

Here we need only the relationship, which is reciprocal : in 

78 



PARAMETERS 6/7 

an absolute system, a change of stability can only be due to 
change of value of a parameter, and change of value of a parameter 
causes a change in stability. 



Reference 

Girden, E., and Culler, E. Conditioned responses in curarized striate 
muscle in dogs. Journal of Comparative Psychology, 23, 2G1 ; 1937. 



79 



CHAPTER 7 



Step-Functions 



7/1. Sometimes the behaviour of a variable (or parameter) can 
be described without reference to the cause of the behaviour : if 
we say a variable or system is a ' simple harmonic oscillator ' 
the meaning of the phrase is well understood. Here we shall be 
more interested in the extent to which a variable displays con- 
stancy. Four types may be distinguished, and are illustrated in 




-^v 



D 



TIME— ►- 

Figure 7/1/1 : Types of behaviour of a variable : A, the full-function 
B, the part-function ; C, the step-function ; D, the null-function. 



Fig. 7/1/1. (A) The full-function has no finite interval of con- 
stancy ; many common physical variables are of this type : the 
height of the barometer, for instance. (B) The part-function has 
finite intervals of change and finite intervals of constancy; it 
will be considered more fully in S. 14/12. (C) The step-function 
has finite intervals of constancy separated by instantaneous jumps. 

80 



STEP-FUNCTIONS 7/2 

And, to complete the set, we need (D) the null-function, which 
shows no change over the whole period of observation. The four 
types obviously include all the possibilities, except for mixed 
forms. The variables of Fig. 2/10/1 will be found to be part-, 
full-, step-, and null-, functions respectively. 

In all cases the type-property is assumed to hold only over 
the period of observation : what might happen at other times 
is irrelevant. 

Sometimes physical entities cannot readily be allotted their 
type. Thus, a steady musical note may be considered either as 
unvarying in intensity, and therefore a null-function, or as 
represented by particles of air which move continuously, and 
therefore a full-function. In all such cases the confusion is at 
once removed if one ceases to think of the real physical object 
with its manifold properties, and selects that variable in which 
one happens to be interested. 

7/2. Step-functions occur abundantly in nature, though the 
very simplicity of their properties tends to keep them incon- 
spicuous. ' Things in motion sooner catch the eye than what 
not stirs '. The following examples approximate to the step- 
function, and show its ubiquity : 

(1) The electric switch has an electrical resistance which 

remains constant except when it changes by a sudden 
jump. 

(2) The electrical resistance of a fuse similarly stays at a low 

value for a time and then suddenly changes to a very 
high value. 

(3) The viscosity of water, measured as the temperature 

passes 0° C, changes similarly. 

(4) If a piece of rubber is stretched, the pull it exerts is approxi- 

mately proportional to its length. The constant of 
proportionality has a definite constant value unless the 
elastic is stretched so far that it breaks. When this 
happens the constant of proportionality suddenly 
becomes zero, i.e. it changes as a step-function. 

(5) If a trajectory is drawn through the air, a few feet above the 

ground and parallel to it, the resistance it encounters as it 
meets various objects varies in step-function form. 
81 



7/3 DESIGN FOR A BRAIN 

(6) A stone, falling through the air into a pond and to the 

bottom, would meet resistances varying similarly. 

(7) The temperature of a match when it is struck changes in 

step-function form. 

(8) If strong acid is added in a steady stream to an un- 

buffered alkaline solution, the pH changes in approxi- 
mately step-function form. 

(9) If alcohol is added slowly with mixing to an aqueous 

solution of protein, the amount of protein precipitated 
changes in approximately step-function form. 

(10) As the pH is changed, the amount of adsorbed substance 

often changes in approximately step-function form. 

(11) By quantum principles, many atomic and molecular 

variables change in step-function form. 

(12) The blood flow through the ductus arteriosus, when ob- 

served over an interval including the animal's birth, 
changes in step-function form. 

(13) The sex-hormone content of the blood changes in step- 

function form as an animal passes puberty. 

(14) Any variable which acts only in ' all or none ' degree shows 

this form of behaviour if each degree is sustained over a 
finite interval. 

7/3. Few variables other than the atomic can change instan- 
taneously ; a more minute examination shows that the change 
is really continuous : the fusing of an electric wire, the closing of a 
switch, and the snapping of a piece of elastic. But if the event 
occurs in a system whose changes are appreciable only over some 
longer time, it may be treated without serious error as if it oc- 
curred instantaneously. Thus, if x — tanh t, it will give a graph 
like A in Figure 7/3/1 if viewed over the interval from t = — 2 
to t = +2. But if viewed over the interval from t — — 40 to 
t = -|- 40, it would give a graph like B, and would approximate 
to the step-function form. 

In any experiment, some ' order ' of the time-scale is always 
assumed, for the investigation never records both the very quick 
and the very slow. Thus to study a bee's honey-gathering flights, 
the observer records its movements. But he ignores the movement 
caused by each stroke of the wing : such movements are ignored 
as being too rapid. Equally, over an hour's experiment he ignores 

82 



STEP-FUNCTIONS 



7/4 



the fact that the bee at the end of the hour is a little older than it 

was at the beginning : this change is ignored as being too slow. 

Such changes are eliminated by being treated as if they had 

their limiting values. If a single rapid change occurs, it is 



B 



Time — *• 

Figure 7/3/1 : The same change viewed : (A) over one interval 
of time, (B) overman interval twenty times as long. 

treated as instantaneous. If a rapid oscillation occurs, the 
variable is given its average value. If the change is very slow, 
the variable is assumed to be constant. In this way the concept 
of ' step-function ' may legitimately be applied to real changes 
which are known to be not quite of this form. 



7/4. Behaviour of step-function form is likely to be seen when- 
ever we observe a ' machine ' whose component parts are fast- 
acting. Thus, if we casually alter the settings of an unknown 
electronic machine we are not unlikely to observe, from time to 
time, sudden changes of step-function form, the suddenness being 
due to the speed with which the machine changes. 

A reason can be given most simply by reference to Figure 4/3/1 . 
Suppose that the curvature of the surface is controlled by a para- 
meter which makes A rise and B fall. If the ball is resting at A, 
the parameter's first change will make no difference to the ball's 
lateral position, for it will continue to rest at A (though with 
lessened reaction if displaced.). As the parameter is changed 
further, the ball will continue to remain at A until A and B are 
level. Still the ball will make no movement. But if the para- 
meter goes on changing and A rises above B, and if gravitation is 



7/5' DESIGN FOR A BRAIN 

intense and the ball fast-moving, then the ball will suddenly move 
to B. And here it will remain, however high A becomes and 
however low B. So, if the parameter changes steadily, the 
lateral position of the ball will tend to step-function form, ap- 
proximating more closely as the passage of the ball for a given 
degree of slope becomes swifter. 

The possibility need not be examined further, for no exact 
deductions will be drawn from it. The section is intended only 
to show that step-functions occur not uncommonly when the 
system under observation contains fast-acting components. The 
subject will be referred to again in S. 10/5. 

Critical states 

7/5. In any absolute system, the behaviour of a variable at any 
instant depends on the values which the variable and the others 
have at that instant (S. 2/15). If one of the variables behaves as 
a step-function the rule still applies : whether the variable remains 
constant or undergoes a change is determined both by the value 
of the variable and by the values of the other variables. So, 
given an absolute system with a step-function at a particular value, 
all the states with the step-function at that value can be divided 
into two classes : those whose occurrence does and those whose 
occurrence does not lead to a change in the step-function's value. 
The former are its critical states : should one of them occur, the 
step-function will change value. The critical state of an electric 
fuse is the number of amperes which will cause it to blow. The 
critical state of the ' constant of proportionality ' of an elastic 
strand is the length at which it breaks. 

An example from physiology is provided by the urinary bladder 
when it has developed an automatic intermittently-emptying 
action after spinal section. The bladder fills steadily with urine, 
while at first the spinal centres for micturition remain inactive. 
When the volume of urine exceeds a certain value the centres 
become active and urine is passed. When the volume falls below 
a certain value, the centre becomes inactive and the bladder refills. 
A graph of the two variables would resemble Figure 7/5/1 . The 
two- variable system is absolute, for it has the field of Figure 7/5/2. 
The variable y is approximately a step-function. When it is at 0, 
its critical state is x = X 2 , y = 0, for the occurrence of this state 

84 




7/6 



TIME-*- 



Figure 7/5/1 : Diagram of the changes in x, volume of urine in the bladder, 
and y, activity in the centre for micturition, when automatic action has 
been established after spinal section. 




'1 ~2 

Figure 7/5/2 : Field of the changes shown in Figure 7/5/1, 



determines a jump from to F. When it is at Y, its critical 
state is x = X v y = Y, for the occurrence of this state determines 
a jump from Y to 0. 

7/6. A common, though despised, property of every machine is 
that it may 4 break '. This event is in no sense unnatural, since 
it must follow the basic laws of physics and chemistry and is 
therefore predictable from its immediately preceding state. In 
general, when a machine ' breaks ' the representative point has met 
some critical state, and the corresponding step-function has changed 
value. 

As is well known, almost any machine or physical system will 
break if its variables are driven far enough away from their usual 
values. Thus, machines with moving parts, if driven ever faster, 
will break mechanically ; electrical apparatus, if subjected to 
ever higher voltages or currents, will break in insulation ; 
machines made too hot will melt — if made too cold they may 
encounter other sudden changes, such as the condensation which 
stops a steam-engine from working below 100° C. ; in chemical 
dynamics, increasing concentrations may meet saturation, or may 
cause precipitation of proteins. 

Although there is no rigorous law, there is nevertheless a wide- 

85 G 



7/7 DESIGN FOR A BRAIN 

spread tendency for systems to show changes of step-function 
form if their variables are driven far from some usual value. 
Later (S. 10/2) it will be suggested that the nervous system is not 
exceptional in this respect. 

Systems containing full- and null-functions 
7/7. We shall now consider the properties shown by absolute 
systems that contain step-functions. But the discussion will be 
clearer and simpler if we first examine some simpler systems. 

Suppose we have an absolute system composed wholly of full- 
functions and we ignore one of the variables. Every experimenter 
knows only too well what happens : the behaviour of the system 
becomes unpredictable. Every experimenter has spent time 
trying to make unpredictable experiments predictable ; he does 
it by identifying the unknown variable. The unknown variable 
may be scientifically trivial, like a loose screw, or important, like 
a co-enzyme in a metabolic system ; but in either case, he cannot 
establish a definite form of behaviour until he has identified and 
either controlled or observed the unknown variable. To ignore a 
/wZZ-function in an absolute system is to render the remainder non- 
absolute, so that no characteristic form of behaviour can be 
established. 

On the other hand, an absolute system which includes null- 
functions may have the null-functions removed from it, or other 
null-functions added to it, and the new system will still be absolute. 
(The alteration is done, of course, not by interfering physically 
with the 4 machine ', but by changing the list of variables.) Thus, if 
the two-variable system of the pendulum (S. 6/3) is absolute, and 
if the length of the pendulum stays constant once it is adjusted, 
then the system composed of the three variables : 

(1) length of pendulum 

(2) angular deviation 

(3) angular velocity) 

is also absolute. A formal proof is given in S. 21/4, but it follows 
readily from the definitions. (The reader should first verify that 
every null-function is itself an absolute system.) Conversely, if 
three variables A, B, N, are found to form an absolute system, 
and N is a null-function, then the system composed of A and B 
is absolute. 

Unlike the full-function, then, the null-function may be 

86 



STET-F UNCTIONS 



7/8 



omitted from a system, for its omission leaves the remainder still 
producing predictable behaviour. 

Systems containing step-functions 

7/8. Suppose that we have a system with three variables, 
A, B, S ; that it has been tested and found absolute ; that A 
and B are full-functions ; and that S is a step-function. (Vari- 
ables A and B, as in S. 21/3, will be referred to as main variables.) 
The phase-space of this system will resemble that of Figure 7/8/1 
(a possible field has been sketched in). The phase-space no longer 
fills all three dimensions, but as S can take only discrete values, 
here assumed for simplicity to be a pair, the phase-space is 
restricted to two planes normal to S, each plane corresponding to a 
particular value of S. A and B being full-functions, the represen- 
tative point will move on curves in each plane, describing a line of 
behaviour such as that drawn more heavily in the Figure. When 




Figure 7/8/1 : Field of an absolute system of three variables, of which 
S is a step-function. The states from C to C are the critical states of 
the step-function. 

the line of behaviour meets the row of critical states at C — C, S 
jumps to its other value, and the representative point continues 
along the heavily marked line in the upper plane. In such a field 
the movement of the representative point is everywhere state- 
determined, for the number of lines from any point never exceeds 
one. 

If, still dealing with the same real c machine ', we ignore S, 
and repeatedly form the field of the system composed of A and B, 
S being free to take sometimes one value and sometimes the other, 
we shall find that we get sometimes a field like I in Figure 7/8/2, 
and sometimes a field like II, the one or the other appearing ac- 
cording to the value that S happens to have at the time. 

87 



7/9 DESIGN FOR A BRAIN 

The behaviour of the system A B, in its apparent possession of 
two fields, should be compared with that of the system described 
in S. 6/3, where the use of two parameter- values also caused the 
appearance of two fields. But in the earlier case the change of 
the field was caused by the arbitrary action of the experimenter, 
who forced the parameter to change value, while in this case the 
change of the field of A B is caused by the inner mechanisms of the 
4 machine ' itself. 

The property may now be stated in general terms. Suppose, 
in an absolute system, that some of the variables are step-functions, 
and that these are ignored while the remainder (the main variables) 
are observed on many occasions by having their field constructed. 

I 





B 

Figure 7/8/2 : The two fields of the system composed of A and B. 
P is in the same position in each field. 

Then so long as no step-function changes value during the con- 
struction, the main variables will be found to form an absolute 
system, and to have a definite field. But on different occasions 
different fields may be found. The number of different fields shown 
by the main variables is equal to the number of combinations of 
values provided by the step-functions. 



1J9. These considerations throw light on an old problem in the 
theory of mechanisms. 

Can a ' machine ' be at once determinate and capable of spon- 
taneous change ? The question would be contradictory if posed 
by one person, but it exists in fact because, when talking of living 
organisms, one school maintains that they are strictly determinate 
while another school maintains that they are capable of spon- 
taneous change. Can the schools be reconciled ? 

88 



STEP-FUNCTIONS 7/9 

The presence of step-functions in an absolute system enables 
both schools to be right, provided that those who maintain the 
determination are speaking of the system which comprises all the 
variables, while those who maintain the possibility of spontaneous 
change are speaking of the main variables only. For the whole 
system, which includes the step-functions, is absolute, has one field 
only, and is completely state-determined (like Figure 7/8/1). But 
the system of main variables may show as many different forms of 
behaviour (like Figure 7/8/2, I and II) as the step-functions 
possess combinations of values. And if the step-functions are not 
accessible to observation, the change of the main variables from 
one form of behaviour to another will seem to be spontaneous, for 
no change or state in the main variables can be assigned as its 
cause. 

The argument may seem plausible, but it is stronger than that. 
It may be proved (S. 22/5) that if a 4 machine ', known to be 
completely isolated and therefore absolute, produces several 
characteristic forms of behaviour, i.e. possesses several fields, then 
there must be, interacting with the observed variables and included 
within the c machine ', some step-functions. 



89 



CHAPTER 8 

The Ultrastable System 



8/1. Our problem, stated briefly at the end of Chapter 5, can 
now be stated finally. The type-problem was the kitten whose 
behaviour towards a fire was at first chaotic and unadapted, 
but whose behaviour later became effective and adapted. We 
have recognised (S. 5/8) that the property of being ' adapted ' 
is equivalent to that of having the variables, both of the animal 
and of the environment, so co-ordinated in their actions on one 
another that the whole system is stable. We now know, from 
S. 6/3 and 7/8, that an observed system can change from one 
form of behaviour to another only if parameters have changed 
value. Since we assumed originally that no deus ex machina 
may act on it, the changes in the system must be due to step- 
functions acting within the whole absolute system. Our problem 
therefore takes the final form : Step-functions by their changes in 
value are to change the behaviour of the system ; what can ensure 
that the step functions shall change appropriately ? The answer is 
provided by a principle, relating step-functions and fields, which 
will now be described. 

8/2. In S. 7/8 it was shown that when a step-function changes 
value, the field of the main variables is changed. The process 
was illustrated in Figures 7/8/1 and 7/8/2. This is the action 
of step-function on field. 

8/3. There is also a reciprocal action. Fields differ in the rela- 
tion of their lines of behaviour to the critical states. Thus, if 
a representative point is started at random in the region to the 
left of the critical states in Figure 8/3/1, the proportion which 
will encounter critical states is, in I — 1, in II — 0, and in III — 
about a half. So, given a distribution of critical states and a 
distribution of initial states, a change of field will, in general, 

90 



THE ULTRASTABLE SYSTEM 



8/5 




[II 





Figure 8/3/1 : Three fields. The critical states are dotted. 

change the proportion of representative points encountering 
critical states. 



The ultrastable system 

8/4. The two factors of the two preceding sections will now be 
found to generate a process, for each in turn evokes the other's 
action. The process is most clearly shown in what I shall call 
an ultrastable system : one that is absolute and contains step- 
functions in a sufficiently large number for us to be able to ignore 
the finiteness of the number. Consider the field of its main 
variables after the representative point has been released from 
some state. If the field leads the point to a critical state, a 
step-function will change value and the field will be changed. 
If the new field again leads the point to a critical state, again 
a step-function will change and again the field will be changed ; 
and so on. The two factors, then, generate a process. 



8/5. Clearly, for the process to come to an end it is necessary 
and sufficient that the new field should be of a form that does 
not lead the representative point to a critical state. (Such a 
field will be called terminal.) But the process may also be de- 
scribed in rather different words : if we watch the main variables 
only, we shall see field after field being rejected until one is 
retained : the process is selective towards fields. 

As this selectivity is of the highest importance for the solution 
of our problem, the principle of ultrastability will be stated 
formally : an ultrastable system acts selectively towards the fields 
of the main variables, rejecting those that lead the representative 
point to a critical state but retaining those that do not. 

This principle is the tool we have been seeking ; the previous 

91 



8/6 DESIGN FOR A BRAIN 

chapters have been working towards it : the later chapters will 
develop it. 

8/6. In the previous sections, the critical states of the step- 
functions were unrestricted in position ; but such freedom does 
not correspond with what is found in biological systems (S. 9/8), 
so we will examine the behaviour of an ultrastable system whose 
critical states are so sited that they surround a definite region 
in the main-variables' phase-space. (At first we shall assume 
that the main variables are all full-functions, though the defini- 
tion makes no such restriction. Later (S. 11/8) we shall examine 
other possibilities.) 

8/7. The simplest way to demonstrate the properties of this 
system is by an example. Suppose there are only two main 
variables, A and B, and the critical states of all the step-functions 



HI 




Figure 8/7/1 



Changes of field in an ultrastable system. The critical 
states are dotted. 



are distributed as the dots in Figure 8/7/1. Suppose the first 
field is that of Figure 8/7/1 (I), and that the system is started 
with the representative point at X. The line of behaviour from 

92 



THE ULTRASTABLE SYSTEM 8/8 

X is not stable in the region, and the representative point follows 
the line to the boundary. Here (F) it meets a critical state 
and a step-function changes value ; a new field, perhaps like II, 
arises. The representative point is now at Y, and the line from 
this point is still unstable in regard to the region. The point 
follows the line of behaviour, meets a critical state at Z, and 
causes a change of a step-function : a new field (III) arises. 
The point is at Z, and the field includes a stable resting state, 
but from Z the line leads further out of the region. So another 
critical state is met, another step-function changes value, and 
a new field (IV) arises. In this field, the line of behaviour from 
Z is stable with regard to the region. So the representative 
point moves to the resting state and stops there. No further 
critical states are met, no further step-functions change value, 
and therefore no further changes of field take place. From now 
on, if the field of the main variables is examined, it will be found 
to be stable. // the critical states surround a region, the ultra- 
stable system is selective for fields that are stable within the region. 
(This statement is not rigorously true, for a little ingenuity 
can devise fields of bizarre type which are not stable but which 
are, under the present conditions, terminal. A fully rigorous 
statement would be too clumsy for use in the next few chapters ; 
but the difficulty is only temporary, for S. 13/4 introduces some 
practical factors which will make the statement practically true.) 

The Homeostat 

8/8. So far the discussion of step-functions and of ultrastability 
has been purely logical. In order to provide an objective and 
independent test of the reasoning, a machine has been built 
according to the definition of the ultrastable system. This 
section will describe the machine and will show how its behaviour 
compares with the prediction of the previous section. 

The homeostat (Figure 8/8/1) consists of four units, each of 
which carries on top a pivoted magnet (Figure 8/8/2, M in 
Figure 8/8/3). The angular deviations of the four magnets from 
the central positions provide the four main variables. 

Its construction will be described in stages. Each unit emits 
a D.C. output proportional to the deviation of its magnet from 
the central position. The output is controlled in the following 

93 



8/8 



DESIGN FOR A BRAIN 



•:*iia 



JbSfe* 




Figure 8/8/1 : The homeostat. Each unit carries on top a magnet and 
coil such as that shown in Figure 8/8/2. Of the controls on the front 
panel, those of the upper row control the potentiometers, those of the 
middle row the commutators, and those of the lower row the switches S 
of Figure 8/8/3. 




Figure 8/8/2 : Typical magnet (just visible), coil, pivot, vane, and water 
potentiometer with electrodes at each end. The coil is quadruple, con- 
sisting of A, B, C and D of Figure 8/8/3. 



94 



THE ULTRASTABLE SYSTEM 



8/8 



way. In front of each magnet is a trough of water ; electrodes 
at each end provide a potential gradient. The magnet carries 
a wire which dips into the water, picks up a potential depending 
on the position of the magnet, and sends it to the grid of the 
triode. J provides the anode-potential at 150 V., while // is at 
180 V. ; so £ carries a constant current. If the grid-potential 
allows just this current to pass through the valve, then no current 
will flow through the output. But if the valve passes more, or 
less, current than this, the output circuit will carry the difference 



s©tS 



ZP 



K<*& 



^ 



u _^ 



^ 




&^s, 



Figure 8/8/3 : Wiring diagram of one unit. (The letters are explained 

in the text.) 



in one direction or the other. So after E is adjusted, the output 
is approximately proportional to il/'s deviation from its central 
position. 

Next, the units are joined together so that each sends its 
output to the other three ; and thereby each receives an input 
from each of the other three. 

These inputs act on the unit's magnet through the coils A, 

B, and C, so that the torque on the magnet is approximately 
proportional to the algebraic sum of the currents in A, B, and 

C. (D also affects M as a self -feedback.) But before each 
input current reaches its coil, it passes through a commutator 

95 



8/8 DESIGN FOR A BRAIN 

(J£), which determines the polarity of entry to the coil, and 
through a potentiometer (P), which determines what fraction of 
the input shall reach the coil. 

As soon as the system is switched on, the magnets are moved 
by the currents from the other units, but these movements change 
the currents, which modify the movements, and so on. It may 
be shown (S. 19/11) that if there is sufficient viscosity in the 
troughs, the four-variable system of the magnet-positions is 
approximately absolute. To this system the commutators and 
potentiometers act as parameters. 

When these parameters are given a definite set of values, the 
magnets show some definite pattern of behaviour ; for the para- 
meters determine the field, and thus the lines of behaviour. If 
the field is stable, the four magnets move to the central position, 
where they actively resist any attempt to displace them. If 
displaced, a co-ordinated activity brings them back to the centre. 
Other parameter-settings may, however, give instability ; in 
which case a c runaway ' occurs and the magnets diverge from 
the central positions with increasing velocity. 

So far, the system of four variables has been shown to be 
dynamic, to have Figure 4/12/1 (A) as its diagram of immediate 
effects, and to be absolute. Its field depends on the thirty-two 
parameters X and P. It is not yet ultrastable. But the inputs, 
instead of being controlled by parameters set by hand, can be 
sent by the switches S through similar components arranged on 
a uniselector (or ' stepping-switch ') U. The values of the com- 
ponents in U were deliberately randomised by taking the actual 
numerical values from Fisher and Yates' Table of Random 
Numbers. Once built on to the uniselectors, the values of these 
parameters are determined at any moment by the positions of 
the uniselectors. Twenty-five positions on each of four uni- 
selectors (one to each unit) provide 390,625 combinations of 
parameter- values. In addition, the coil G of each uniselector is 
energised when, and only when, the magnet M diverges far from 
the central position ; for only at extreme divergence does the 
output-current reach a value sufficient to energise the relay F 
which closes the coil-circuit. A separate device, not shown, 
interrupts the coil-circuit regularly, making the uniselector move 
from position to position as long as F is energised. 

The system is now ultrastable ; its correspondence with the 

96 



THE ULTRASTABLE SYSTEM 8/8 

definition will be shown in each of the three requirements. 
Firstly, the whole system, now of eight variables (four of the 
magnet- deviations and four of the uniselector-positions), is abso- 
lute, because the values of the eight variables are sufficient to 
determine its behaviour. Secondly, the variables may be divided 
into main variables (the four magnet-deviations), and step-func- 
tions (the variables controlled by the uniselector-positions). 
Thirdly, as the uniselectors provide an almost endless supply of 
step-function values (though not all different) we do not have to 
consider the possibility that the supply of step-function changes 
will come to an end. In addition, the critical states (those 
magnet-deviations at which the relay closes) are all sited at about 
a 45° deviation ; so in the phase-space of the main variables they 
form a 4 cube ' around the origin. 

It should be noticed that if only one, two, or three of the 
units are used, the resulting system is still ultrastable. It will 
have one, two, or three main variables respectively, but the critical 
states will be unaltered in position. 



Time 



Figure 8/8/4 : Behaviour of one unit fed back into itself through a uniselector. 
The upper line records the position of the magnet, whose side-to-side 
movements are recorded as up and down. The lower line (U) shows 
a cross-stroke whenever the uniselector moves to a new position. The 
first movement at each D was forced by the operator, who pushed the 
magnet to one side to make it demonstrate the response. 

Its ultrastability can now be demonstrated. First, for sim- 
plicity, is shown a single unit arranged to feed back into itself 
through a single uniselector coil such as A, D being shorted out. 
In such a case the occurrence of the first negative setting on 
the uniselector will give stability. Figure 8/8/4 shows a typical 
tracing. At first the step-functions gave a stable field to the 
single main variable, and the downward part of D l9 caused by 
the operator deflecting the magnet, is promptly corrected by the 
system, the magnet returning to its central position. At R v 

97 



8/8 DESIGN FOR A BRAIN 

the operator reversed the polarity of the output-input junction, 
making the system unstable (S. 20/7). As a result, a runaway 
developed, and the magnet passed the critical state (shown by 
the dotted line). As a result the uniselector changed value. As 
it happened, the first new value provided a field which was 
stable, so the magnet returned to its central position. At D 2 , 
a displacement showed that the system was now stable (though 
the return after R x demonstrated it too). 

At R 2 the polarity of the join was reversed again. The value 
on the uniselector was now no longer suitable, the field was 
unstable, and a runaway occurred. This time three uniselector 
positions provided three fields which were all unstable : all were 





Time 



Figure 8/8/5 : Two units (1 and 2) interacting. (Details as in Fig. 8/8/4.) 

rejected. But the fourth was stable, the magnet returned to 
the centre, no further uniselector changes occurred, and the 
single main variable had a stable field. At D 3 its stability was 
again demonstrated. 

Figure 8/8/5 shows another experiment, this time with two 
units interacting. The diagram of immediate effects was 1 <± 2 ; 
the effect 1 — >■ 2 was hand-controlled, and 2 — > 1 was uniselector- 
controlled. At first the step-function values combined to give 
stability, shown by the responses to D v At R v reversal of the 
commutator by hand rendered the system unstable, a runaway 
occurred, and the variables transgressed the critical states. The 
uniselector in Unit 1 changed position and, as it happened, gave 
at its first trial a stable field. It will be noticed that whereas 
before R x the upstroke of D 1 in 2 caused an upstroke in 1, it 

98 



THE ULTRASTABLE SYSTEM 8/9 

caused a down stroke in 1 after R v showing that the action 2 — ► 1 
had been reversed by Jhe uniselector. This reversal compensated 
for the reversal of 1 — > 2 caused at R v 

At R 2 the whole process was repeated. This time three uni- 
selector changes were required before stability was restored. A 
comparison of the effect of Z) 3 on 1 with that of D 2 shows that 
compensation has occurred again. 

The homeostat can thus demonstrate the elementary facts of 
ultrastability. 

8/9. In what way does an ultrastable system differ from an 
ordinary stable system ? 

In one sense the two systems are similar. Each is assumed 
absolute, and if therefore we form the field of all its variables, 
each will have one permanent field. Given a region, every line 
of behaviour is permanently stable or unstable (see Figure 7/8/1). 
Viewed in this way, the two systems show no essential difference. 
But if we compare the variables of the stable system with only 
the main variables of the ultrastable, then an obvious difference 
appears : the field of the stable system is single and permanent, 
but in the ultrastable system the phase-space of the main vari- 
ables shows a succession of transient fields concluded by a terminal 
field which is always stable, The distinction in actual behaviour 
can best be shown by an example. The automatic pilot is a 
device which, amongst other actions, keeps the aeroplane hori- 
zontal. It must therefore be connected to the ailerons in such 
a way [that when the plane rolls to the right, its output [must 
act on them so as to roll the plane to the left. If properly joined, 
the whole system is stable and self-correcting : it can now fly 
safely through turbulent air, for though it will roll frequently, 
it will always come back to the level. The homeostat, if joined 
in this way, would tend to do the same. (Though not well 
suited, it would, in principle, if given a gyroscope, be able to 
correct roll.) 

So far they show no difference ; but connect the ailerons in 
reverse and compare them. The automatic pilot would act, 
after a small disturbance, to increase the roll, and would persist 
in its wrong action to the very end. The homeostat, however, 
would persist in its wrong action only until the increasing devia- 
tion made the step-functions start changing. On the occurrence 

99 



8/10 DESIGN FOR A BRAIN 

of the first suitable new value, the homeostat would act to stabilise 
instead of to overthrow ; it would return the plane to the hori- 
zontal ; and it would then be ordinarily self-correcting for dis- 
turbances. 

There is therefore some justification for the name ' ultrastable ' ; 
for if the main variables are assembled so as to make their field 
unstable, the ultrastable system will change this field till it is 
stable. The degree of stability shown is therefore of an order 
higher than that of the system with a single field. 

Another difference can be seen by considering the number of 
factors which need adjustment or specification in order to achieve 
stability. Less adjustment is needed if the system is ultrastable. 
Thus an automatic pilot must be joined to the ailerons with care, 
but an ultrastable pilot could safely be joined to the ailerons at 
random. Again, a linear system of n variables, to be made stable, 
needs the simultaneous adjustment of at least n parameters 
(S. 20/11, Ex. 3). If n is, say, a thousand, then at least a thou- 
sand parameters must be correctly adjusted if stability is to be 
achieved. But an ultrastable system with a thousand main 
variables needs, to achieve stability, the specification of about 
six factors ; for this is approximately the number of independent 
items in the specification of the system (S. 9/9). A large system, 
then, can be made stable with much less detailed specification 
if it is made ultrastable. 

8/10. In S. 6/2 it was shown that every dynamic system is 
acted on by an indefinitely large number of parameters, many of 
which are taken for granted, for they are always given well- 
understood 4 obvious ' values. Thus, in mechanical systems it 
is taken for granted, unless specially mentioned, that the bodies 
carry a zero electrostatic charge ; in physiological experiments, 
that the tissues, unless specially mentioned, contain no unusual 
drug ; in biological experiments, that the animal, unless specially 
mentioned, is in good health. All these parameters, however, 
are effective in that, had their values been different, the variables 
would not have followed the same line of behaviour. Clearly 
the field of an absolute system depends not only on those para- 
meters which have been fixed individually and specifically, but 
on all the great number which have been fixed incidentally. 
Now the ultrastable system proceeds to a terminal field which 

100 



THE ULTRASTABLE SYSTEM 



8/10 



is stable in conjunction with all the system's parameter-values 
(and it is clear by the principle of ultrastability that this must be 
so, for whether the parameters are at their ' usual ' values or 
not is irrelevant). The ultrastable system will therefore always 
produce a set of step-function values which is so related to the 
particular set of parameter-values that, in conjunction with them, 
the system is stable. If the parameters have unusual values, 






U 



m 



Time 



Figure 8/10/1 : Three units interacting. At J, units 1 and 2 were con- 
strained to move together. New step-function values were found which 
produced stability. These values give stability in conjunction with the 
constraint, for when it is removed, at R, the system becomes unstable. 



the step-functions will also finish with values that are compen- 
satingly unusual. To the casual observer this adjustment of the 
step-function values to the parameter- values may be surprising ; 
we, however, can see that it is inevitable. 

The fact is demonstrable on the homeostat. After the machine 
was completed, some ' unusual ' complications were imposed on 
it (' unusual ' in the sense that they were not thought of till 
the machine had been built), and the machine was then tested 
to see how it would succeed in finding a stable field when 
affected by the peculiar complications. One such test was 

101 H 



8/10 DESIGN FOR A BRAIN 

made by joining the front two magnets by a light glass fibre 
so that they had to move together. Figure 8/10/1 shows a 
typical record of the changes. Three units were joined together 
and were at first stable, as shown by the response when the 
operator displaced magnet 1 at D v At J, the magnets of 1 and 
2 were joined so that they could move only together. The result 
of the constraint in this case was to make the system unstable. 
But the instability evoked step-function changes, and a new 
terminal field was found. This was, of course, stable, as was 
shown by its response to the displacement, made by the operator, 
at D 2 . But it should be noticed that the new set of step-function 
values was adjusted to, or 'took notice of, the constraint and, 
in fact, used it in the maintenance of stability ; for when, at R, 
the operator gently lifted the fibre away the system became 
unstable. 

References 

Ashby, W. Ross. Design for a brain. Electronic Engineering, 20, 379 ; 1948. 
Idem. The cerebral mechanisms of intelligent behaviour, in Perspectives in 

Neuropsychiatry, edited D. Richter. London, 1950. 
Idem. Can a mechanical chess-player outplay its designer? British Journal 

for the Philosophy of Science, 3, 44 ; 1952. 
Fisher, R. A., and Yates, F. Statistical tables. Edinburgh, 1943. 



102 



CHAPTER 9 



Ultrastability in the 
Living Organism 



9/1. The principle of ultrastability has so far been treated as 
a principle in its own right, true or false without reference to 
possible applications. This separation has prevented the possi- 
bility of a circular argument ; but the time for its application 
has now come. I propose, therefore, the thesis that the living 
organism uses the principle of ultrastability as an automatic 
means of ensuring the adaptiveness of its learned behaviour. 
At first I shall cite only facts in its favour, leaving all major 
criticisms to Chapter 11. We shall have, of course, to assume 
that the animal, and particularly the nervous system, contains 
the necessary variables behaving as step-functions : whether this 
assumption is reasonable will be discussed in the next chapter. 

Examples of adaptive, learned behaviour are so multitudinous 
that it will be quite impossible for me to discuss, or even to 
mention, the majority of them. I can only select a few as 
typical and leave the reader to make the necessary modifications 
in other cases. 

The best introduction is not an example of learned behaviour, 
but Jennings' classic description of the reactions of Stentor, a 
single-celled pond animalcule. I shall quote him at length : 

4 Let us now examine the behaviour [of Stentor] under 
conditions which are harmless when acting for a short time, 
but which, when continued, do^ interfere with the normal 
functions. Such conditions may be produced by bringing a 
large quantity of fine particles, such as India ink or carmine, 
by means of a capillary pipette, into the water currents 
which are carried to the disc of Stentor. 

1 Under these conditions the normal movements are at 
first not changed. The particles of carmine are taken into 
the pouch and into the mouth, whence they pass into the 
internal protoplasm. If the cloud of particles is very dense, 

103 



9/1 DESIGN FOR A BRAIN 

or if it is accompanied by a slight chemical stimulus, as is 
usually the case with carmine grains, this behaviour lasts 
but a short time ; then a definite reaction supervenes. The 
animal bends to one side ... It thus as a rule avoids the 
cloud of particles, unless the latter is very large. This 
simple method of reaction turns out to be more effective 
in getting rid of stimuli of all sorts than might be expected. 
If the first reaction is not successful, it is usually repeated 
one or more times . . . 

4 If the repeated turning toward one side does not relieve 
the animal, so that the particles of carmine continue to come 
in a dense cloud, another reaction is tried. The ciliary 
movement is suddenly reversed in direction, so that the 
particles against the disc and in the pouch are thrown off. 
The water current is driven away from the disc instead of 
toward it. This lasts but an instant, then the current is 
continued in the usual way. If the particles continue to 
come, the reversal is repeated two or three times in rapid 
succession. If this fails to relieve the organism, the next 
reaction — contraction — usually supervenes. 

4 Sometimes the reversal of the current takes place before 
the turning away described first ; but usually the two 
reactions are tried in the order we have given. 

4 If the Stentor does not get rid of the stimulation in either 
of the ways just described, it contracts into its tube. In 
this way it of course escapes the stimulation completely, 
but at the expense of suspending its activity and losing all 
opportunity to obtain food. The animal usually remains 
in the tube about half a minute, then extends. When its 
body has reached about two-thirds its original length, the 
ciliary disc begins to unfold and the cilia to act, causing 
currents of water to reach the disc, as before. 

4 We have now reached a specially interesting point in 
the experiment. Suppose that the water currents again 
bring the carmine grains. The stimulus and all the external 
conditions are the same as they were at the beginning. 
Will the Stentor behave as it did at the beginning ? Will 
it at first not react, then bend to one side, then reverse the 
current, then contract, passing anew through the whole 
series of reactions ? Or shall we find that it has become 
changed by the experiences it has passed through, so that 
it will now contract again into its tube as soon as stimulated ? 

4 We find the latter to be the case. As soon as the car- 
mine again reaches its disc, it at once contracts again. This 
may be repeated many times, as often as the particles come 
to the disc, for ten or fifteen minutes. Now the animal 
after each contraction stays a little longer in the tube than 
it did at first. Finally it ceases to extend, but contracts 

104 



ULTRASTABILITY IN THE LIVING ORGANISM 9/1 

repeatedly and violently while still enclosed in its tube. In 
this way the attachment of its foot to the object on which 
it is situated is broken and the animal is free. Now it 
leaves its tube and swims away. In leaving the tube it may 
swim forward out of the anterior end of the tube ; but if 
this brings it into the region of the cloud of carmine, it 
often forces its way backwards through the substance of 
the tube, and thus gains the outside. Here it swims away, 
to form a new tube elsewhere. 

1 . . . the changes in behaviour may be summed up as 
follows : 

(1) No reaction at first ; the organism continues its normal 

activities for a time. 

(2) Then a slight reaction by turning into a new position. 

(3) ... a momentary reversal of the ciliary current . . . 

(4) . . . the animal breaks off its normal activity com- 

pletely by contracting strongly . . . 

(5) ... it abandons its tube . . . ' 

The behaviour of Stentor bears a close resemblance to the 
behaviour of an ultrastable system. The physical correspon- 
dences necessary would be as follows : — Stentor and its environ- 
ment constitute an absolute system by S. 3/9 ; for Jennings, 
having set the carmine flowing, interferes no further. They 
consequently correspond to the whole ultrastable system, which 
is also absolute by the definition of S. 8/4. The observable 
(here : visible) variables of Stentor and its environment corre- 
spond to the main variables of the ultrastable system. In Stentor 
are assumed to be variables which behave like, and correspond 
to, the step -functions of the ultrastable system. The critical 
states of the organism's step-functions surround the region of 
the normal values of the organism's essential variables so that 
its step-functions change value if the essential variables diverge 
widely from their usual, normal values. These critical states 
must be nearer to the normal value than the extreme limits of 
the essential variables, for these critical states must be reached 
before the essential variables reach the extreme limits compatible 
with life. 

Now compare the behaviour of the ultrastable system, de- 
scribed in S. 8/7, with the behaviour of organisms like Stentor, 
epitomised by Jennings in these words : 

1 Anything injurious to the organism causes changes in 
its behaviour. These changes subject the organism to new 

105 



9/2 DESIGN FOR A BRAIN 

conditions. As long as the injurious condition continues, 
the changes of behaviour continue. The first change of 
behaviour may not be regulatory [what I call ' adaptive '], 
nor the second, nor the third, nor the tenth. But if the 
changes continue, subjecting the organism successively to 
all possible different conditions, a condition will finally be 
reached that relieves the organism from the injurious action, 
provided such a condition exists. Thereupon the changes 
in behaviour cease and the organism remains in the favourable 
situation.' 

The resemblance between my statement and his is obvious. 
Jennings grasped the fundamental fact that aimless change can 
lead to adaptation provided that some active process rejects the 
bad and retains the good. He did not, however, give any physical 
(i.e. non-vital) reason why this selection should occur. He records 
only that it does occur, and that its occurrence is sufficient to 
account for adaptation at the primitive level. 

The first example therefore suggests that, provided we are 
willing to assume that Stentor contains step-functions which (a) 
affect Stentor* s behaviour, and (b) have critical states that are 
encountered before the essential variables reach their extreme 
limits, Stentor may well achieve its final adaptation by using 
the automatic process of ultrastability. 

9/2. The next example includes more complicating factors but 
the main features are clear. Mowrer put a rat into a box with 
a grilled metal floor. The grill could be electrified so as to give 
shocks to the rat's paws. Inside the box was a pedal which, 
if depressed, at once stopped the shocks. 

When a rat was put into the box and the electric stimulation 
started, the rat would produce various undirected activities such 
as jumping, running, squealing, biting at the grill, and random 
thrashing about. Sooner or later it would depress the pedal 
and stop the shocks. After the tenth trial, the application of 
the shock would usually cause the rat to go straight to the pedal 
and depress it. These, briefly, are the observed facts. 

Consider the internal linkages in this system. We can suffi- 
ciently specify what is happening by using six variables, or sets 
of variables : those shown in the box-diagram below. By con- 
sidering the known actions of part on part in the real system 
we can construct the diagram of immediate effects. Thus, the 

106 



ULTRASTABILITY IN THE LIVING ORGANISM 



9/2 



excitations in the motor cortex certainly control the rat's bodily 
movements, and such excitations have no direct effect on any of 
the other five groups of variables ; so we can insert arrow 1, 
and know that no other arrow leaves that box. (The single 
arrow, of course, represents a complex channel.) Similarly, the 
other arrows of the diagram can be inserted. Some of the 
arrows, e.g. 2 and 4, represent a linkage in which there is not 



Events in 


6 


Events in 


sensory cortex 




motor cortex 


5 


k 




1 


Excitation 
in skin 




Position of 
limbs 


> 
4 


^ 




2 


Voltage 




Position 


on | 


shrill 


3 


of pedal 



a positive physical action all the time ; but here, in accordance 
with S. 2/3, we regard them as permanently linked though some- 
times acting at zero degree. 

Having completed the diagram, we notice that it forms a 
functional circuit. The system is complete and isolated, and 
may therefore be treated as absolute. To apply our thesis, we 
assume that the cerebral part, represented by the boxes around 
arrow 6, contains step -functions whose critical states will be 
transgressed if stimuli of more than physiological intensity are 
sent to the brain. 

We now regard the system as straightforwardly ultrastable, 
and predict what its behaviour must be. It is started, by hypo- 
thesis, from an initial state at which the voltage is high. This 
being so, the excitation at the skin and in the brain will be high. 
At first the pattern of impulses sent to the muscles does not 
cause that pedal movement which would lower the voltage on 
the grill. These high excitations in the brain will cause some 
step-functions to change value, thus causing different patterns 
of body movement to occur. The step-functions act directly 
only at stage 6, but changes there will (S. 14/11) affect the field 

107 



9/3 DESIGN FOR A BRAIN 

of all six groups of main variables. These changes of field will 
continue to occur as long as the high excitation in the brain 
persists. They will cease when, and only when, the linkages at 
stage 6 transform an excitation of skin receptors into such a 
bodily movement as will cause, through the pedal, a reduction 
in the excitation of the skin receptors ; for only such linkages 
can stop further encounters with critical states. The system 
that is, will change until there occurs a stable field. The stability 
will be shown by an increase in the voltage on the grill leading 
to changes through skin, brain, muscles, and pedal that have 
the effect of opposing the increase in voltage. The stability, in 
addition, has the property that it keeps the essential variables 
within physiological limits ; for by it the rat is protected from 
electrical injury, and the nervous system from exhaustion. 

It will be noted that although action 3 has no direct connec- 
tion, either visually in the real apparatus or functionally in the 
diagram of immediate effects, with the site of the changes at 6, 
yet the latter become adapted to the nature of the action at 3. 
The subject was discussed in S. 5/15. 

This example shows, therefore, that if the rat and its environ- 
ment formed an ultrastable system and acted purely automati- 
cally, they would have gone through the same changes as were 
observed by Mowrer. 

9/3. The two examples have taken a known fact of animal 
behaviour and shown its resemblance to the behaviour of the 
ultrastable system. Equally, the behaviour of the homeostat, 
a system known to be ultrastable, shows some resemblance to 
that of a rudimentary nervous system. The tracings of Figures 
8/8/4 and 8/8/5 show its elementary power of adaptation. In 
Figure 8/8/5 the reversal at R x might be regarded as the action 
of an experimenter who changed the conditions so that the i aim ' 
(stability and homeostasis) could be achieved only if the ' organ- 
ism ' (Unit 1) reversed its action. Such a reversal might be 
forced on a rat who, having learned a maze whose right fork 
led to food, was transferred to a maze where food was to be 
found only down the left fork. The homeostat, as Figure 8/8/5 
shows, develops a reversed action in Unit 1, and this reversal 
may be compared with the reversal which is usually found to 
occur in the rat's behaviour. 

108 



ULTRASTABILITY IN THE LIVING ORGANISM 9/3 

A more elaborate reaction by the homeostat is shown in 
Figure 9/3/1. 



1 



\r 



V 




tt 



Time 



Figure 9/3/1 : Three units interacting. At R the effect 
of 2 on 3 was reversed in polarity. 

The machine was arranged so that its diagram of immediate 
effects was 



>3 



The effect 3 — > 1 was set permanently so that a movement of 
3 made 1 move in the opposite direction. The action 1 — > 2 
was uniselector-controlled, and 2 — > 3 hand-controlled. When 
the tracing commenced, the actions 1 — > 2 and 2 — > 3 were 
demonstrated by the downward movement, forced by the operator, 
of 1 at S t : 2 followed 1 downward (similar movement), and 3 
followed 2 downward (similar movement). 3 then forced 1 up- 
ward, opposed the original movement, and produced stability. 
At R, the hand-control (2 — > 3) was reversed, so that 2 now 
forced 3 to move in the opposite direction to itself. This change 
set up a vicious circle and destroyed the stability ; but uniselector 
changes occurred until the stability was restored. A forced 
downward movement of 1, at S 2 , demonstrated the regained 
stability. 

109 



9/4 DESIGN FOR A BRAIN 

The tracing, however, deserves closer study. The action 2 — ► 3 
was reversed at R, and the responses of 2 and 3 at S 2 demon- 
strate this reversal ; for while at S ± they moved similarly, at S 2 
they moved oppositely. Again, a comparison of the uniselector- 
controlled action 1 — > 2 before and after R shows that whereas 
beforehand 2 moved similarly to 1, afterwards it moved oppo- 
sitely. The reversal in 2 — > 3, caused by the operator, thus 
evoked a reversal in 1 — > 2 controlled by the uniselector. The 
second reversal is compensatory to the first. 

The nervous system provides many illustrations of such a 
series of events : first the established reaction, then an altera- 
tion made in the environment by the experimenter, and finally 
a reorganisation within the nervous system, compensating for 
the experimental alteration. The homeostat can thus show, in 
elementary form, this power of self-reorganisation. 



The necessity of ultrastability 

9/4. In the previous sections a few simple examples have sug- 
gested that the adaptation of the living organism may be due 
to ultrastability. But the argument has not excluded the possi- 
bility that other theories might fit the facts equally well. I shall 
now give, therefore, evidence to show that ultrastability is not 
merely plausible but necessary : the organism must be ultra- 
stable. 

First the primary assumptions : they are such as few scientists 
would doubt. It is assumed that the organism and its environ- 
ment form an absolute system, and that the organism sometimes 
changes from one regular way of behaving to another. The 
crucial question is whether we can prove that the organism's 
mechanism must contain step-functions. In S. 22/5 is given 
such a proof, stated in mathematical form ; but its theme is 
simple and can be stated in plain words. 

Suppose a ' machine ' or experiment behaves regularly in one 
way, and then suddenly changes to behaving in another way, 
again regularly. Suppose, for instance, a pharmacologist, test- 
ing the effect of a new drug on the frog's heart, finds at every 
test all through one day that it causes the pulse-rate to lessen. 
Next morning, taking records of the effect, he finds at every 

110 



ULTRASTABILITY IN THE LIVING ORGANISM 9/4 

attempt that it causes the pulse-rate to increase. He will almost 
certainly ask himself 4 What has changed ? ' 

Such facts provide valid evidence that some variable has 
changed value. I need not elaborate the logic for no experi- 
menter would question it. What has been sometimes overlooked 
though, is that we are also entitled to draw the deduction that 
the variable, being as it is an effective factor towards the system, 
must, throughout the previous day, have remained constant ; 
for otherwise the reactions observed during the day could not 
have been regular. For the same reason, it must also have been 
constant throughout the next morning. And further, the two 
constant values cannot have been equal, for then the hearts' 
behaviours would not have been changed. Assembling these 
inferences, we deduce that the variable must have behaved as 
a step-function. Exactly the same argument, applied to the 
changes of behaviour shown by Jennings' Stentor, leads to the 
deduction that within the organism there must have been vari- 
ables behaving as step-functions. 

Is there any escape from this conclusion ? It rests primarily 
on the simple thesis that a determinate system does not, if started 
from identical states, do one thing on one day and something 
else on another day. There seems to be no escape if we assume 
that the systems we are discussing are determinate. Suppose, 
then, that we abandon the assumption of determinism and allow 
indeterminism of atomic type to affect heart, Stentor, or brain 
to an observable extent. This would allow us to explain the 
' causeless ' overnight change ; but then we would be unable to 
explain the regularity throughout the previous day and the next 
morning. It seems there is no escape that way. Again, we 
could, with a little ingenuity, construct a hypothesis that the 
pharmacologist's experiment was affected by a small group of 
variables, whose joint action produced the observed result but 
not one of which was a step-function ; and it might be claimed 
that the theorem had been shown -false. But this is really no 
exception, for we are not concerned with what variables ' are ' 
but with how they behave, and in particular with how they 
behave towards the system in question. If a group of variables 
behaves towards the system as a step-function, then it is a step- 
function ; for the ' step-function ' is defined primarily as a form 
of behaviour, not as a thing. 

Ill 



9/5 DESIGN FOR A BRAIN 

Once it is agreed that a system, such as that of Mowrer's rat, 
contains step-functions, then all it needs is that they should 
not be few for the system to be admitted as ultrastable. 

After this, we can examine the qualifications that were added 
when considering Stentor as an ultrastable system. Are they, 
too, necessary ? Not with the assumptions made so far in this 
section, but they become so if we add the postulates that the 
system ' adapts ' in the sense of S. 5/8, and that it does so by 
' trial and error'. In order to be definite about what 'trial and 
error ' implies, here is the concept defined explicitly : 

(1) The organism makes trials only when ' dissatisfied ' or 

4 irritated ' in some way. 

(2) Each trial persists for a finite time. 

(3) While the irritation continues, the succession of trials 

continues. 

(4) The succeeding trial is not specially related to the preced- 

ing, nor better than it, but only different. 

(5) The process stops at the first trial that relieves the irritation. 
The argument goes thus. As each step-function forms part 

of an absolute system, its change must depend on its own and 
on the other variables' values ; there must, therefore, be certain 
states — the critical — at which it changes value. When, in the 
process of adaptation by trial and error, the step-function changes 
value, its critical states must have been encountered ; and since, 
by (1) above, the step-functions change value only when the 
organism is ' dissatisfied ' or ' irritated ', the critical states must 
be so related to the essential variables that only when the organism 
is driven from its normal physiological state does its representa- 
tive point encounter the critical states. This knowledge is suffi- 
cient to place the critical states in the functional sense : they 
must have values intermediate between those of the normal state 
and those of the essential variables' limits. The qualifications 
introduced in S. 9/1 are thus necessary. 

Training 

9/5. The process of ' training ' will now be shown in its relation 
to ultrastability. 

All training involves some use of ' punishment ' or ' reward ', 
and we must translate these concepts into our form. ' Punish- 

112 



ULTRASTABILITY IN THE LIVING ORGANISM 9/5 

ment ' is simple, for it means that some sensory organs or nerve 
endings have been stimulated with an intensity high enough to 
cause step-function changes in the nervous system (S. 7/6 and 
10/2). The concept of c reward ' is more complex. It usually 
involves the supplying of some substance (e.g. food) or condition 
(e.g. escape) whose absence would act as ' punishment '. The 
chief difficulty is that the evidence suggests that the nervous 
system, especially the mammalian, contains intricate and special- 
ised mechanisms which give the animals properties not to be 
deduced from basic principles alone. Thus it has been shown 
that dogs with an oesophageal fistula, deprived of water for some 
hours, would, when offered water, drink approximately the 
quantity that would correct the deprivation, and would then 
stop drinking ; they would stop although no water had entered 
stomach or system. The properties of these mechanisms have 
not yet been fully elucidated ; so training by reward uses 
mechanisms of unknown properties. Here we shall ignore these 
complications. We shall assume that the training is by pain, 
i.e. by some change which threatens to drive the essential vari- 
ables outside their normal limits ; and we shall assume that 
training by reward is not essentially dissimilar. 

It will now be shown that the process of ' training ' necessarily 
implies the existence of feedback. But first the functional rela- 
tionship of the experimenter to the experiment must be made 
clear. 

The experimenter often plays a dual role. He first plans 
the experiment, deciding what rules shall be obeyed during it. 
Then, when these have been fixed, he takes part in the experi- 
ment and obeys these rules. With the first role we are not 
concerned. In the second, however, it is important to note that 
the experimenter is now within the functional machinery of the 
experiment. The truth of this statement can be appreciated 
more readily if his place is taken by an untrained but obedient 
assistant who carries out the instructions blindly ; or better still 
if his place is taken by an apparatus which carries out the pre- 
scribed actions automatically. 

When the whole training is arranged to occur automatically 
the feedback is readily demonstrated if we construct the diagram 
of immediate effects. Thus, a pike in an aquarium was separated 
from some minnows by a sheet of glass ; every time he dashed 

113 



9/5 DESIGN FOR A BRAIN 

at the minnows he struck the glass. The following immediate 
effects can be clearly distinguished : 



Activities in 


1 


Activities in 


motor cortex 




muscles 


4 


2 


Activities in 


Pressure on 


sensory 


cortex 


3 


nose 



The arrow 1 represents the control exerted through spinal cord 
and motor nerves. Effect 2 is discontinuous but none the less 
clear : the experiment implies that some activities led to a high 
pressure on the nose while others led to a zero pressure. Effects 
3 and 4 are the simple neuro-physiological results of pressures 
on the nose. 

Although the diagram has some freedom in the selection of 
variables for naming, the system, regarded as a whole, clearly 
has feedback. 

In other training experiments, the regularity of action 2 
(supplied above by the constant physical properties of glass) 
may be supplied by an assistant who constantly obeys the rules 
laid down by the experimenter. Grindley, for instance, kept a 
guinea-pig in a silent room in which a buzzer was sounded from 
time to time. If and only if its head turned to the right did a 
tray swing out and present it with a piece of carrot ; after a 
few nibbles the carrot was withdrawn and the process repeated. 
Feedback is demonstrably present in this system, for the diagram 
of immediate effects is : 



Activities in 


1 


Position of 


motor cortex 




head 


4 

4 


. 


> 


2 


Activities in 


Amount of 


sensory 


cortex 


3 


carrot p 


resented 



The buzzer, omitted for clarity, comes in as parameter and serves 
merely to call this dynamic system into functional existence ; 
for only when the buzzer sounds does the linkage 2 exist. 



114 



ULTRASTABILITY IN THE LIVING ORGANISM 9/5 

This type of experiment reveals its essential dynamic structure 
more clearly if contrasted with elementary Pavlovian condition- 
ing. In the experiments of Grindley and Pavlov, both use the 
sequences ' . . . buzzer, animal's response, food . . .' In Grindley's 
experiment, the value of the variable 4 food ' depended on the 
animal's response : if the head turned to the left, ' food ' was 
' no carrot ', while if the head turned to the right, ' food ' was 
4 carrot given \ But in Pavlov's experiments the nature of 
every stimulus throughout the session was already determined 
before the session commenced. The Pavlovian experiment, there- 
fore, allows no effect from the variable 4 animal's behaviour ' to 
4 quantity of food given ' ; there is no functional circuit and no 
feedback. 

It may be thought that the distinction (which corresponds to 
that made by Hilgard and Marquis between 4 conditioning ' and 
4 instrumental learning ') is purely verbal. This is not so, for 
the description given above shows that the distinction may be 
made objectively by examining the structure of the experiment. 
Culler et al. performed an experiment in which feedback, at first 
absent, was added at an intermediate stage : as a result, the 
dog's behaviour changed. They gave the dog a shock to the 
leg and sounded a tone. The reaction to the shock was one of 
generalised struggling movements of the body and retraction of 
the leg. After a few sessions the tone produced generalised 
struggling and retraction of the leg. So far there had been no 
feedback ; but now the conditions were changed : the shock was 
given at the tone only if the foot was not raised. As a result 
the dog's behaviour changed : the response rapidly narrowed to 
a simple and precise flexion of the leg. 

It will be seen, therefore, that the 4 training ' situation neces- 
sarily implies that the trainer, or some similar device, is an 
integral part of the whole system, which has feedback : 



Trainer 




Animal 







We shall now suppose this system to be ultrastable, and we 
shall trace its behaviour on this supposition. The step-functions 
are, of course, assumed to be confined to the animal ; both 
because the human trainer may be replaced in some experiments 

115 



9/5 DESIGN FOR A BRAIN 

by a device as simple as a sheet of glass (in the example of the 
pike) ; and because the rules of the training are to be decided in 
advance (as when we decide to punish a house-dog whenever he 
jumps into a chair), and therefore to be invariant throughout 
the process. Suppose then that jumping into a chair always 
results in the dog's sensory receptors being excessively stimulated. 
As an ultrastable system, step-function values which lead to 
jumps into chairs will be followed by stimulations likely to cause 
them to change value. But on the occurrence of a set of step- 
function values leading to a remaining on the ground, excessive 
stimulation will not occur, and the values will remain. (The 
cessation of punishment when the right action occurs is no less 
important in training than its administration after the wrong 
action.) 

The process can be shown on the homeostat. Figure 9/5/1 
provides an example. Three units were joined : 



and to this system was joined a ' trainer ', actually myself, which 
acted on the rule that if the homeostat did not respond to a 
forced movement of 1 by an opposite movement of 2, then the 
trainer would force 3 over to an extreme position. The diagram 
of immediate effects is therefore really 



>T 




Part of the system's feedbacks, it will be noticed, pass through T. 
At S v 1 was moved and 2 moved similarly. This is the * for- 
bidden ' response ; so at D 1} 3 was forced by the trainer to an 
extreme position. Step-functions changed value. At S 2 , the 
homeostat was tested again : again it produced the forbidden 
response ; so at Z) 2 , 3 was again forced to an extreme position. 
At £3, the homeostat was tested again : it moved in the desired 
way, so no further deviation was forced on 3. And at *S 4 and 
$5 the homeostat continued to show the desired reaction. 

116 



ULTRASTABILITY IN THE LIVING ORGANISM 9/6 

From Si onwards, T's behaviour is determinate at every instant ; 
so the system composed of 1, 2, 3, T, and the uniselectors, is 
absolute. 

Another property of the whole system should be noticed. 
When the movement-combination 4 1 and 2 moving similarly ' 
occurs, T is thereby impelled, under the rules of the experiment, 
to force 3 outside the region bounded by the critical states. Of 
any inanimate system which behaved in this way we would 




Figure 9/5/1 : Three units interacting. The downstrokes at S are 
forced by the operator. If 2 responds with a downstroke, the 
trainer drives 3 past its critical surface. 

say, simply, that the line of behaviour from the state at which 
1 and 2 started moving was unstable. So, to say in psychological 
terms that the ' trainer ' has i punished ' the 4 animal ' is equiva- 
lent to saying in our terms that the system has a set of step- 
function values that make it unstable. 

In general, then, we may identify the behaviour of the animal 
in ( training ' with that of the ultrastable system adapting to 
another system of fixed characteristics. 



9/6. A remarkable property of the nervous system is its ability 
to adapt itself to surgical alterations of the bodily structure. 
From the first work of Marina to the recent work of Sperry, such 
experiments have aroused interest and no little surprise. 

Over thirty years ago, Marina severed the attachments of the 
internal and external recti muscles of a monkey's eyeball and 
re-attached them in crossed position so that a contraction of 

117 I 



9/6 DESIGN FOR A BRAIN 

the external rectus would cause the eyeball to turn not outwards 
but inwards. When the wound had healed, he was surprised 
to discover that the two eyeballs still moved together, so that 
binocular vision was preserved. 

More recently Sperry severed the nerves supplying the flexor 
and extensor muscles in the arm of the spider monkey, and re- 
joined them in crossed position. After the nerves had regenerated, 
the animal's arm movements were at first grossly inco-ordinated 
but improved until an essentially normal mode of progression 
was re-established. The two examples are typical of a great 
number of experiments, and will suffice for the discussion. 

In S. 3/12 it was decided that the anatomical criterion for 
dividing the system into ' animal ' and ' environment ' is not 
the only possible : a functional criterion is also possible. Suppose 
a monkey, to get food from a box, has to pull a lever towards 
itself ; if we sever the flexor and extensor muscles of the arm 
and re-attach them in crossed position then, so far as the cerebral 
cortex is concerned, the change is not essentially different from 
that of dismantling the box and re-assembling it so that the 
lever has to be pushed instead of pulled. Spinal cord, peripheral 
nerves, muscles, bones, lever, and box — all are 'environment' 
to the cerebral cortex. A reversal in the cerebral cortex will 
compensate for a reversal in its environment whether in spinal 
cord, muscles, or lever. It seems reasonable, therefore, to expect 
that the cerebral cortex will use the same compensatory process 
whatever the site of reversal. 

I have already shown, in S. 8/10 and in Figure 8/10/1, that 
the ultrastable system arrives at a stability in which the values 
of the step-functions are related to those of the parameters of 
the system, i.e. to the surrounding fixed conditions, and that 
the relation will be achieved whether the parameters have values 
which are * normal ' or are experimentally altered from those 
values. If these conclusions are applied to the experiments of 
Marina and Sperry, the facts receive an explanation, at least in 
outline. To apply the principle of ultrastability we must add 
an assumption that ' binocular vision ' and ' normal progression ' 
have neural correlates such that deviations from binocular vision 
or from normal progression cause an excitation sufficient to cause 
changes of step-function in those cerebral mechanisms that 
determine the actions. (The plausibility of this assumption will 

118 



ULTRASTABILITY IN THE LIVING ORGANISM 9/7 

be discussed in S. 9/8.) Ultrastability will then automatically 
lead to the emergence of behaviour which produces binocular 
vision or normal progression. For this to be produced, the step- 
function values must make appropriate allowance for the par- 
ticular characteristics of the environment, whether ' crossed ' or 
4 uncrossed '. S. 8/10 and Figure 9/3/1 showed that an ultra- 
stable system will make such allowance. The adaptation shown 
by Marina's monkey is therefore homologous with that shown 
by Mowrer's rat, for the same principle is responsible for both. 

9/7. ' Learning ' and ' memory ' are vast subjects, and any 
theory of their mechanisms cannot be accepted until it has been 
tested against all the facts. It is not my intention to propose 
any such theory, since this work confines itself to the problem 
of adaptation. Nevertheless I must indicate briefly the relation 
of this work to the two concepts. 

4 Learning ' and ' memory ' have been given almost as many 
definitions as there are authors to write of them. The concepts 
involve a number of aspects whose interrelations are by no means 
clear ; but the theme is that a past experience has caused some 
change in the organism's behaviour, so that this behaviour is 
different from what it would have been if the experience had not 
occurred. But such a change of behaviour is also shown by a 
motor-car after an accident ; so most psychologists have insisted 
that the two concepts should be restricted to those cases in 
which the later behaviour is better adapted than the earlier. 

The ultrastable system shows in its behaviour something of 
these elementary features of ' learning '. In Figure 9/3/1, for 
instance, the pattern of behaviour produced at S 2 is different 
from the pattern at S v The change has occurred after the 
1 experience ' of the instability at R. And the new field pro- 
duced by the step-function change is better adapted than the 
previous field, for an unstable field has been replaced by a stable. 

An elementary feature of ' memory ' is also shown ; for further 
responses, S 3 , S^ etc. would repeat S%s pattern of behaviour, 
and thereby might be said to show a ' memory ' of the reversal 
at R ; for the later pattern is adapted to the reversal at R, and 
not adapted to the original setting. 

The ultrastable system, then, shows rudimentary ' learning ' 
and ' memory '. The subject is resumed in S. 11/3. 

119 



9/8 



DESIGN FOR A BRAIN 

The control of aim 



9/8. The ultrastable systems discussed so far, though develop- 
ing a variety of fields, have sought a constant goal. The homeo- 
stat sought central positions and the rat sought zero grill-potential. 
In this section will be described some methods by which the 
goal may be varied. 

If the critical states' distribution in the main-variables' phase- 
space is altered by any means whatever, the ultrastable system 



10 



5- 



IO 



2Q 



U 



Figure 9/8/1. 

will be altered in the goal it seeks. For the ultrastable system 
will always develop a field which keeps the representative point 
within the region of the critical states (S. 8/7). Thus if (Figure 
9/8/1) for some reason the critical states moved to surround B 
instead of A, then the terminal field would change from one which 
kept x between and 5 to one which kept x between 15 and 20. 
A related method is illustrated by Figure 9/8/2. An ultra- 
stable system U interacts with a variable A. 
E and R represent the immediate effects which 
U and A have on each other ; they may be 
thought of as A's effectors and receptors. If 
A should have a marked effect on the ultra- 
stable system, the latter will, of course, develop 
a field stabilising A ; at what value will depend 
markedly on the action of R. Suppose, for 
instance, that U has its critical states all at 
values and 10, so that it always selects a field 
stabilising all its main variables between these values. If R 
is such that, if A has some value a, R transmits to U the 
value 5a — 20, then it is easy to see that U will develop a field 
holding A within one unit of the value 5 ; for if the field makes 

120 




A 

Figure 9/8/2. 



ULTRASTABILITY IN THE LIVING ORGANISM 9/8 

A go outside the range 4 to 6, it will make U go outside the 
range to 10, and this will destroy the field. So U becomes 
w 5-seeking '. If the action of R is now changed to transmitting, 
not 5a — 20 but 5a + 5, then U will change fields until it 
holds A within one unit of ; and U is now ' 0-seeking .' So 
anything that controls the b in R = 5a + b controls the ' goal ' 
sought by U. 

As a more practical example, suppose U is mobile and is 
ultrastable, with its critical states set so that it seeks situations 
of high illumination ; such would occur if its critical states 
resembled, in Figure 9/8/1, B rather than A. Suppose too that 
R is a ray of light. If in the path of R we place a red colour- 
filter, then green light will count as ' no light ' and the system 
will actively seek the red places and avoid the green. If now 
we merely replace the red filter by a green, the whole aim of 
its movements will be altered, for it will now seek the green 
places and avoid the red. 

Next, suppose R is a transducer that converts a temperature 
at A into an illumination which it transmits to U. If R is 
arranged so that a high temperature at A is converted into a high 
illumination, then U will become actively goal-seeking for hot 
places. And if the relation within R is reversed, U will seek 
for cold places. Clearly, whatever controls R controls C/'s goal. 

There is therefore in general no difficulty in accounting for 
the fact that a system may seek one goal at one time and another 
goal at another time. 

Sometimes the change, of critical states or of the transducer 
R, may be under the control of a single parameter. When this 
happens we must distinguish two complexities. Suppose the 
parameter can take only two values and the system U is very 
complicated. Then the system is simple in the sense that it 
will seek one of only two goals, and is complicated in the sense 
that the behaviour with which it gets to the goal is complicated. 
That the behaviour is complicated is no proof, or even sugges- 
tion, that the parameter's relations to the system must be com- 
plicated ; for, as was shown in S. 6/3, the number of fields is 
equal to the number of values the parameter can take, and has 
nothing to do with the number of main variables. It is this 
latter that determines, in general, the complexity of the goal- 
seeking behaviour. 

121 



9/9 DESIGN FOR A BRAIN 

These considerations may clarify the relations between the 
change of concentration of a sex-hormone in the blood of a 
mammal and its consequent sexual goal-seeking behaviour. A 
simple alternation between ' present ' and 4 absent ', or between 
two levels with a threshold, would be sufficient to account for 
any degree of complexity in the two behaviours, for the com- 
plexity is not to be related to the hormone-parameter but to the 
nervous system that is affected by it. Since the mammalian 
nervous system is extremely complex, and since it is, at almost 
every point, sensitive to both physical and chemical influences, 
there seems to be no reason to suppose that the directiveness of 
the sex-hormones on the brain's behaviour is essentially different 
from that of any parameter on the system it controls. (That 
the sex-hormones evoke specifically sexual behaviour is, of course, 
explicable by the fact that evolution, through natural selection, 
has constructed specific mechanisms that react to the hormone 
in the specific way.) 

Ultrastability and the gene-pattern 

9/9. In S. 1/9 it was pointed out that although the power of 
adaptation shown by a species ultimately depends on its genetic 
endowment, yet the number of genes is, in the higher animals, 
quite insufficient to specify every detail of the final neuronic 
organisation. It was suggested that in the higher animals, the 
genes must establish function-rules which will look after the 
details automatically. 

As the minimal function-rules have now been provided (S. 8/7) 
it is of interest to examine the specification of the ultrastable 
system to see how many items will have to be specified geneti- 
cally if the ovum is to grow into an ultrastable organism. The 
items are as follows : 

(1) The animal and its environment must form an absolute 

system (S. 3/9) ; 

(2) The system must be actively dynamic ; 

(3) Essential variables must be defined for the species (S. 3/14) ; 

(4) Step-functions are to be provided (S. 8/4) ; 

(5) Their critical states are all to be similar (S. 8/6) ; 

(6) The critical states are to be related in value to the limiting 

values of the essential variables (S. 9/1). 
122 



ULTRA STABILITY IN THE LIVING ORGANISM 9/10 

From these basic rules, an ultrastable system of any size can 
be generated by mere repetition of parts. Thus each critical 
state is to have a value related to the limits of the essential vari- 
ables ; but this requirement applies to all other critical states 
by mere repetition. The repetition needs fewer genes than would 
be necessary for independent specification. 

It is not possible to give an exact estimate of the number of 
genes necessary to determine the development of an ultrastable 
system. But the number of items listed above is only six ; and 
though the number of genes required is probably a larger number, 
it may well be less than the number known to be available. It 
seems, therefore, that the requirement of S. 1/9 has been met 
satisfactorily. 

9/10. If the higher animals are made ultrastable by their genetic 
inheritance, the gene-pattern must have been shaped by natural 
selection. Could an ultrastable system be developed by natural 
selection ? 

Suppose the original organism had no step-functions ; such an 
organism would have a permanent, invariable set of reactions. 
If a mutation should lead to the formation of a single step-func- 
tion whose critical states were such that, when the organism 
became distressed, it changed value before the essential variables 
transgressed their limits, and if the step-function affected in any 
way the reaction between the organism and the environment, 
then such a step-function might increase the organism's chance 
of survival. A single mutation causing a single step-function 
might therefore prove advantageous ; and this advantage, though 
slight, might be sufficient to establish the mutation as a species 
characteristic. Then a second mutation might continue the pro- 
cess. The change from the original system to the ultrastable 
can therefore be made by a long series of small changes, each 
of Avhich improves the chance of survival. The change is thus 
possible under the action of natural selection. 



References 

Culler, E., Finch, G., Girden, E., and Brogden, W. Measurements of 
acuity by the conditioned-response technique. Journal of General 
Psychology, 12, 223 ; 1935. 

123 



9/10 DESIGN FOR A BRAIN 

Grindley, G. C. The formation of a simple habit in guinea-pigs. British 

Journal of Psychology, 23, 127 ; 1932-3. 
Hilgard, E. R., and Marquis, D. G. Conditioning and learning. New 

York, 1940. 
Marina, A. Die Relationen des Palaeencephalons (Edinger) sind nicht fix. 

Neurologisches Centralblatt, 34, 338 ; 1915. 
Mowrer, O. H. An experimental analogue of ' regression ' with incidental 

observations on ' reaction-formation '. Journal of Abnormal and Social 

Psychology, 35, 56 ; 1940. 
Sperry, R. W. Effect of crossing nerves to antagonistic limb muscles in 

the monkey. Archives of Neurology and Psychiatry, 58, 452 ; 1947. 



124 



CHAPTER 10 



Step-Functions in the 
Living Organism 



10/1. In S. 9/4 the existence of step-functions in the living 
organism was deduced from the observed facts. But so far 
nothing has been said, other than S. 7/6, about their physio- 
logical nature. What evidence is there of a more practical nature 
to support this deduction and to provide further details ? 

Direct evidence of the existence of step-functions in the living 
organism is almost entirely lacking. What evidence exists will 
be reviewed in this chapter. But the lack of evidence does not, 
of course, prove that such variables do not occur, for no one, so 
far as I am aware, has made a systematic search for them. 
Several reasons have contributed to this neglect. Their signi- 
ficance has not been appreciated, so if they have been mentioned 
in the literature they were probably mentioned only casually; 
and since they show a behaviour bordering on total immobility, 
they would usually have been regarded as uninteresting, and 
may not have been recorded even when observed. It is to be 
hoped that the recognition of the fundamental part which they 
play in the processes of adaptation, of integration, and of co- 
ordination, may lead to a fuller knowledge of their actual nature. 
' The anatomical localisation ', said Claude Bernard, c is often 
revealed first through the analysis of the physiological process.' 
Here I can do no more than to indicate some possibilities. 

10/2. Every cell contains many variables that might change in 
a way approximating to the step-function form, especially if the 
time of observation is long compared with the average time of 
cellular events. Monomolecular films, protein solutions, enzyme 
systems, concentrations of hydrogen and other ions, oxidation- 
reduction potentials, adsorbed layers, and many other constituents 
or processes might behave as step-functions. 

125 



10/3 DESIGN FOR A BRAIN 

If the cell is sufficiently sensitive to be affected by changes of 
atomic size, then such changes would usually be of step-function 
form, for they could change only by a quantum jump. But this 
source of step-functions is probably unavailable, for changes of 
this size may be too indeterminate for the production of the 
regular and reproducible behaviour considered here (S. 1/10). 

Round the neuron, and especially round its dendrons and axons, 
there is a sensitive membrane that might provide step-functions, 
though the membrane is probably wholly employed in the trans- 
mission of the action potential. Nerve ' fibrils ' have been des- 
cribed for many years, though the possibility that they are an arte- 
fact cannot yet be excluded. If they are real their extreme delicacy 
of structure suggests that they might behave as step-functions. 

The delicacy everywhere evident in the nervous system has 
often been remarked. This delicacy must surely imply the 
existence of step-functions ; for the property of being w delicate ' 
can mean little other than 4 easily broken ' ; and it was observed 
in S. 7/6 that the phenomenon of something ' breaking ' is the 
expression of a step-function changing value. Though the argu- 
ment is largely verbal, it gives some justification for the opinion 
that step-functions are by no means unlikely in the nervous system. 

' The idea of a steady, continuous development ', said 
Jacques Loeb, ' is inconsistent with the general physical 
qualities of protoplasm or colloidal material. The colloidal 
substances in our protoplasm possess critical points. . . . 
The colloids change their state very easily, and a number of 
conditions . . . are able to bring about a change in their 
state. Such material lends itself very readily to a discon- 
tinuous series of changes.' 

10/3. Another source of step-functions would be provided if 
neurons were amoeboid, so that their processes could make or 
break contact with other cells. 

That nerve-cells are amoeboid in tissue-culture has been known 
since the first observations of Harrison. When nerve-tissue 
from chick-embryo is grown in clotted plasma, filaments grow 
outwards at about 0-05 mm. per hour. The filament terminates 
in an expanded end, about 15 x 25 fx in size, which is actively 
amoeboid, continually throwing out processes as though explor- 
ing the medium around. Levi studied tissue-cultures by micro- 
dissection, so that individual cells could be stimulated. He found 

126 



STEP-FUNCTIONS IN THE LIVING ORGANISM 10/5 

that a nerve-cell, touched with the needle-point, would sometimes 
throw out processes by amoeboid movement. 

The conditions of tissue-culture are somewhat abnormal, and 
artefacts are common ; but this objection cannot be raised against 
the work of Speidel, who observed nerve-fibres growing into the 
living tadpole's tail. The ends of the fibres, like those in the 
tissue-culture, were actively amoeboid. Later he observed the 
effects of metrazol in the same way : there occurred an active 
retraction and, later, re-extension. More recently Carey and 
others have studied the motor end-plate. They found that it, 
too, is amoeboid, for it contracted to a ball after physical injury. 

To react to a stimulus by amoeboid movement is perhaps the 
most ancient of reactions. Reasons have been given in S. 9/1 
and 9/4 suggesting that adaptation by step-functions is as old as 
protoplasm itself. So the hypothesis that neurons are amoeboid 
assumes only that they have never lost their original property. 
It seems possible, therefore, that step-functions might be provided 
in this way. 

10/4. A variable which can take only the two values ' all ' or 
4 nothing ' obviously provides a step-function. It may not always 
conform to the definition of a step-function, for its change is not 
always sustained ; but such variables may well provide changes 



B 



w -Time-*- - Time — 

Figure 10/4/1. 

which appear elsewhere in step-function form. Such would hap- 
pen, for instance, if the change of the variable X (Figure 10/4/1 A) 
resulted in some accumulative change Y, which would vary as in 
B. Variables like X could therefore readily yield step-functions. 



10/5. Step-functions could also be provided by groups of neurons 
acting as a whole. 

Lorente de No has provided abundant histological evidence that 

127 



10/6 



DESIGN FOR A BRAIN 



neurons form not only chains but circuits. Figure 10/5/1 is taken 
from one of his papers. Such circuits are so common that he has 
enunciated a c Law of Reciprocity of Connexions ' : ' if a cell- 
complex A sends fibres to cell or cell-complex B, then B also sends 
fibres to A, either direct or by means of one internuncial neuron '. 
A simple circuit, if excited, would tend either to sink back to 
zero excitation, if the amplification-factor was less than unity, 
or to rise to maximal excitation if it was greater than unity. 
Such a circuit tends to maintain only two degrees of activity : 




Figure 10/5/1 : Neurons and their connections in the trigeminal 
reflex arc. (Semi-diagrammatic ; from Lorente de N6.) 

the inactive and the maximal. Its activity will therefore be of 
step-function form if the time taken by the chain to build up to 
maximal excitation can be neglected. Its critical states would 
be the smallest excitation capable of starting it to full activity, 
and the smallest inhibition capable of stopping it. McCulloch 
has referred to such circuits as ' endromes ' and has studied some 
of their properties. The reader will notice that the ' endrome ' 
exemplifies the principle of S. 7/4. 

10/6. The definition of the ultrastable system might suggest 
that an almost infinite number of step-functions is necessary if 
the system is not to keep repeating itself ; and the reader may 
wonder whether the nervous system can supply so large a number. 
In fact the number required is not large. The reason can be 
shown most simply by a numerical illustration. 

If a step-function can take two values it can provide two fields 
for the main variables (Figure 7/8/1). If another step-function 
with two values is added, the total combinations of value are four, 
and each combination will, in general, produce its own field 
(S. 21/1). So if there are n step-functions, each capable of taking 

128 



STEP-FUNCTIONS IN THE LIVING ORGANISM 10/6 

two values, the total number of fields available will be 2 n . This 
number would have to be lessened in practical cases for practical 
reasons, but even if it is only approximate, it still illustrates the 
main fact : the number of fields is moderate when n is moderate, 
but rapidly becomes exceedingly large when n increases. Ten 
step-functions, for instance, will provide over a thousand fields, 
while twenty step-functions will provide over a million. The 
number of fields soon becomes astronomic. 

The following imaginary example emphasizes the relation be- 
tween the number of fields and the number of step-functions 
necessary to provide them. If a man used fields at the rate of 
ten a second day and night during his whole life of seventy years, 
and if no field was ever repeated, how many two-valued step- 
functions would be necessary to provide them ? Would the 
reader like to guess ? The answer is that thirty-five would be 
ample ! Quantitatively, of course, the calculation is useless ; but 
it shows clearly that the number of step-functions can be far less 
than the number of fields provided. So if the human nervous 
system produces a very large number of fields, we need not deduce 
that it must have a very large number of step-functions. 

References 

Carey, E. J., Massopust, L. C, Zeit, W., Haushalter, E., and Schmitz, J. 
Studies of ameboid motion and secretion of motor end plates : V, Experi- 
mental pathologic effects of traumatic shock on motor end plates in 
skeletal muscle. Journal of Neuropathology and experimental Neurology, 
4, 134 ; 1945. 

Harrison, R. G. Observations on the living developing nerve fiber. Pro- 
ceedings of the Society for Experimental Biology and Medicine, 4, 140 ; 
1906-7. 

Levi, G. Ricerche sperimentali sovra elementi nervosi sviluppati ' in vitro \ 
Archiv fiir experimented Zellforschung, 2, 244 ; 1925-6. 

Lorente de No, R. Vestibulo-ocular reflex arc. Archives of Neurology and 
Psychiatry, 30, 245 ; 1933. 

McCulloch, W. S. A heterarchy of values determined by the topology of 
nervous nets. Bulletin of mathematical Biophysics, 7, 89 ; 1945. 

Speidel, C. C. Studies of living nerves ; activities of ameboid growth 
cones, sheath cells, and myelin segments as revealed by prolonged observa- 
tion of individual nerve fibers in frog tadpoles. American Journal of 
Anatomy, 52, 1 ; 1933. 

Idem. Studies of living nerves ; VI, Effects of metrazol on tissues of frog 
tadpoles with special reference to the injury and recovery of individual 
nerve fibers. Proceedings of the American Philosophical Society, 83, 
349 : 1940. 



129 



CHAPTER 11 

Fully Connected Systems 



11/1. In the preceding chapters all major criticisms were post- 
poned : the time has now come to admit that the simple ultra- 
stable system, as represented by, say, the homeostat, is by no 
means infallible in its attempts at adaptation. 

But before we conclude that its failures condemn it, we must 
be clear about our aim. The designer of a new giant calculating 
machine and we in this book might both be described as trying 
to design a ' mechanical brain '. But the aims of the two 
designers are very different. The designer of the calculator wants 
something that will carry out a task of specified type, and he 
usually wants it to do the work better than the living brain can 
do it. Whether the machine uses methods anything like those 
used by the living brain is to him a side-issue. My aim, on the 
other hand, is simply to copy the living brain. In particular, 
if the living brain fails in certain characteristic ways, then I want 
my artificial brain to fail too ; for such failure would be valid 
evidence that the model was a true copy. With this in mind, 
it will be found that some of the ultras table system's failures 
in adaptation occur in situations that are well known to be just 
those in which living organisms also are apt to fail. 

(1) If an ultrastable system's critical surfaces are not disposed 
in proper relation to the limits of the essential variables (S. 9/1), 
the system may seek an inappropriate goal or may fail to take 
corrective action when the essential variables are dangerously 
near their limits. 

In animals, though we cannot yet say much about their critical 
states, we can observe failures of adaptation that may well be due 
to a defect of this type. Thus, though animals usually react 
defensively to poisons like strychnine — for it has an intensely 
bitter taste, stimulates the taste buds strongly, and is spat out 
— they are characteristically defenceless against a tasteless or 
odourless poison : precisely because it stimulates no nerve-fibre 

130 



FULLY CONNECTED SYSTEMS 11/1 

excessively and causes no deviation from the routine of chewing and 
swallowing. 

An even more dramatic example, showing how defenceless is 
the living organism if pain has not its normal effect of causing 
behaviour to change, is given by those children who congenitally 
lack the normal self-protective reflexes. Boyd and Nie have 
recently described such a case : a girl, aged 7, who seemed healthy 
and normal in all respects except that she was quite insensitive 
to pain. Even before she was a year old her parents noticed 
that she did not cry when injured. At one year of age her arm was 
noticed to be crooked : X-rays showed a recent fracture-disloca- 
tion. The child had made no complaint, nor did she show any 
sign of pain when the fragments were re-set without an anaesthetic. 
Three months later the same injury occurred to her right elbow. 
At the seaside she crawled on the rocks until her hands and knees 
were torn and denuded of skin. At home her mother on several 
occasions smelt burning flesh and found the child leaning uncon- 
cernedly against the hot stove. 

It seems, then, that if an imperfectly formed ultrastable system 
is, under certain conditions, defenceless, so may be an imperfectly 
formed living organism. 

(2) Even if the ultrastable system is suitably arranged — if the 
critical states are encountered before the essential variables reach 
their extreme limits — it usually cannot adapt to an environment 
that behaves with sudden discontinuities. In the earlier examples 
of the homeostat's successful adaptations the actions were always 
arranged to be continuous ; but suppose the homeostat had con- 
trolled a relay which was usually unchanging but which, if the 
homeostat passed through some arbitrarily selected state, would 
suddenly release a powerful spring that would drag the magnets 
away from their ' optimal ' central positions : the homeostat, if 
it happened to approach the special state, would take no step 
to avoid it and would blindly evoke the 'lethal' action. The 
homeostat's method for achieving adaptation is thus essentially 
useless when its environment contains such ' lethal ' discontinuities. 

The living organism, however, is also apt to fail with just the 
same type of environment. The pike that collided with the 
glass plate while chasing minnows failed at first to avoid collision 
precisely because of the suddenness of the transition from not 
seeing clear glass to feeling the impact on its nose. This flaw 

131 



11/1 DESIGN FOR A BRAIN 

in the living organism's defences has, in fact, long been known 
and made use of by the hunter. The stalking cat's movements 
are such as will maintain as long as possible, for the prey, 
the appearance of a peaceful landscape, to be changed with 
the utmost possible suddenness into one of mortal threat. In the 
whole process the suddenness is essential. Consider too the 
essential features of any successful trap ; and the necessity, in 
poisoning vermin, of ensuring that the first dose is lethal. 

If, then, the ultrastable system usually fails when attempting to 
adapt to an environment with sudden discontinuities, so too does 
the living organism. 

(3) Another weakness shown by the ultrastable system's method 
is that success is dependent on the system's using a suitable 
period of delay between each trial. Thus, the system shown in 
Fig. 8/7/1 must persist in Trial IV long enough for the repre- 
sentative point to get away from the region of the critical states. 
Both extremes of delay may be fatal : too hurried a change from 
trial to trial may not allow time for i success ' to declare itself ; 
and too prolonged a testing of a wrong trial may allow serious 
damage to occur. Up to now I have said nothing of this necessity 
for delay between one trial and the next, but there is no doubt 
that it is an essential part of the ultrastable system's method 
of adaptation. Thus the homeostat needed a device, not shown 
in Fig. 8/8/3, for allowing the uniselectors to move only at about 
every 2-3 seconds. 

In animals, little is known scientifically about the optima for 
such delays. But there can be little doubt that on many occa- 
sions living organisms have missed success either by abandoning 
a trial too quickly, or by persisting too long with a trial that 
was actually useless. 

The same difficulty, then, seems to confront both ultrastable 
system and living organism. 

(4) If we grade an ultrastable system's environments according 
to the difficulty they present, we shall find that at the ' easy ' 
end are those that consist of a few variables, independent of each 
other, and that at the ' difficult ' end are those that contain many 
variables richly cross-linked to form a complex whole. 

The living organism, too, would classify environments in 
essentially the same way. Not only does common experience 
show this, but the construction and use of ' intelligence tests ' 

132 



FULLY CONNECTED SYSTEMS 11/4 

has shown in endless ways that the easy problem is the one 
whose components are few and independent, while the difficult 
problem is the one with many components that form a complex 
whole. So when confronted with environments of various ' diffi- 
culties ', the ultras table system and the living organism are likely 
to fail together. 

It seems, then, that the ultrastable system's modes of failure 
support, rather than discredit, its claim to resemble the living 
brain. 

11/2. Now we can turn to those features in which the simple ultra - 
stable system, as represented by the homeostat, differs markedly 
from the brain of the living organism. One obvious difference 
is shown by the record of Figure 8/8/4, in which the homeostat 
made four attempts at finding a terminal field. After its first 
three trials its success was zero ; then, after its next trial, its 
success was complete. The homeostat can show no gradation in 
success, though this is almost universally observable in the living 
organism : day by day a puppy becomes steadier on its legs ; 
year by year a child improves its education. 

11/3. A second difference is seen in their powers of conservation. 
If the homeostat adapts to an environment A and then to an 
environment B, and is then returned to A again, it has no adapta- 
tion immediately ready, for its old adaptation was destroyed in 
the readjustments to B ; it does not even start with a tendency 
to adapt more quickly than before : its second adaptation to 
A takes place as though its first adaptation had never occurred. 
This, of course, is not the case in living organisms, except perhaps 
in the extremely primitive : a child, by learning what two times 
three is, does not thereby destroy its acquired knowledge of what 
is two times two. 

11/4. Although the homeostat, in adapting to B, usually de- 
stroys its adaptation to A, this is not the case necessarily, and 
we should notice a property, inherent in the ultrastable system, 
that might enable it to adapt to more than one environment. 
It will be described partly for its intrinsic interest, as it will be 
referred to later, and partly to show that it is insufficient to 
remove the main difficulty. 

133 K 



11/4 DESIGN FOR A BRAIN 

Let the homeostat be arranged so that it is partly under uni- 
selector-, and partly under hand-, control. Let it be started so 
that it works as an ultrastable system. Select a commutator 
switch, and from time to time reverse its polarity. This reversal 
provides the system with the equivalent of two environments 
which alternate. We can now predict that it will be selective for 
fields that give adaptation to both environments. For consider 
what field can be terminal : a field that is terminal for only one of 
the parameter- values will be lost when the parameter next changes; 
but the first field terminal for both will be retained. Figure 
11/4/1 illustrates the process. At R 19 R 2 , jR 3 , and R± the hand- 

Jj J, Rj * 3 R4 



V 



lA A. 

* H- 



X h 3l- 



Time 



Figure 11/4/1 : Record of homeostat's behaviour when a commutator H 
was reversed from time to time (at the R's). The first set of uniselector 
values which gave stability for both commutator positions was terminal. 

controlled commutator H was reversed. At first the change of 
value caused a change of field, shown at A. But the second 
uniselector position happened to provide a field which gave 
stability with both values of H. So afterwards, the changes of 
H no longer caused step-function changes. The responses to the 
displacements Z), forced by the operator, show that the system 
is stable for both values of //. The slight but distinct difference 
in the behaviour after D at the two values of H show that the 
two fields are different. 

The ultrastable system is, therefore, selective for step-function 
values which give stability for both values of an alternating 
parameter. 

11/5. Such a process would occur in a biological system if an 
animal had to adapt by one internal arrangement to two environ- 

1.34 



FULLY CONNECTED SYSTEMS 11/6 

ments which affected the animal alternately. Such alternations 
do occur. A cat, for instance, must learn to catch mice, which 
tend to run towards corners, and birds, which tend to fly upwards ; 
and the diving birds alternate between aerial and submarine 
environments. Were the bird's nervous system like the homeo- 
stat, step-function changes would occur until there arose a set of 
values giving behaviour suitable to both environments, and this 
set would then be terminal. That such a set is not impossible is 
shown by the snake's mode of progression, which is suitable in 
both undergrowth and water. 

11/6. But it is easily seen that the process cannot answer the 
problems of this chapter. First, the process shows, contrary 
to requirement, no gradation : when there occurs a set of step- 
function values terminal for both environments, the animal be- 
comes adapted ; prior to that it was unadapted. The second 
reason is that any extensive adaptation in this way is very 
improbable. 

This brings us to the most serious of the difficulties. A suc- 
cessful trial, or a terminal field, is useful for adaptation only if it 
occurs within some reasonable time : success at the millionth trial 
is equivalent to failure. Consequently, the principle of ultra - 
stability, while it guarantees that a field of a certain type will 
be retained, guarantees much less than it seems to. If the delay 
in reaching success were slight, a general increase in the system's 
velocity of action might give sufficient compensation ; but in 
fact the delay is likely to exceed the utmost possible compensa- 
tion. For definiteness, take a numerical example. Suppose that 
in some ultrastable system each field has a one in ten chance 
of being stable with any given environment, and that the chances 
are independent. Then the chance of a field being stable to two 
environments will be one in a hundred, and to N environments 
will be one in 10 N . The time that a system takes on the average 
to find a stable field is proportional to the reciprocal of the prob- 
ability (S. 23/2). Suppose that when N = 1 the average time 
t taken to find a terminal field is 1 second, then 

t = ^.10 Y seconds. 

Try the effect of different values of N. Three environments will 
require about a minute and a half. This might be tolerable. 

135 



11/7 DESIGN FOR A BRAIN 

But if N is twenty the time becomes 3,200,000,000 centuries, which 
for our purpose, is equivalent to ' never '. Other examples, 
though quantitatively different, would lead to the same general 
conclusion : when the number of environments is more than a few, 
the time taken by this method to find a field stable to all exceeds 
the allowable. Evidently our brains do not use this method : 
success by it is too improbable. 

11/7. In the previous section we regarded the animal as having 
to adapt to a variety of environments, but we can also regard 
them as constituting a single 4 total ' environment. This makes 
the number of variables in the system increase. What will be 
the effect of this increase on the time taken to find a terminal 
field ? For instance, could the homeostat adapt if it consisted 
of a hundred units instead of four ? The question cannot be 
ignored, for the human brain contains about 10,000,000,000 
nerve-cells, and to this we must add the number of variables in its 
environment. What is the chance that a field should be terminal 
when it occurs in a system with this number of variables ? 

If the system worked as a magnified homeostat then, although 
exact calculation is impossible, the evidence, reviewed in S. 20/12, 
is sufficient to show that, for practical purposes, there is no chance 
at all. If we were like homeostats, waiting till one field gave us, 
at a stroke, all our adult adaptation, we would wait for ever. 
But the infant does not wait for ever ; on the contrary, the 
probability that he will develop a full adult adaptation within 
twenty years is near to unity. Some extra factor must therefore 
be added if the large ultrastable system is to get adapted within 
a reasonable time. 

11/8. It may seem that we have now proved that the whole 
solution must be wrong. But if we re-trace the argument, we find 
that to some extent the difficulty has been unnecessarily magnified. 
From S. 8/6 onwards we assumed for convenience of discussion 
that every main variable was in full dynamic interaction with 
every other main variable, so that every change in every variable 
at once affected every other variable. This gives a system that is 
extremely active and that unquestionably acts as a whole, not as 
a collection of small parts acting independently. As an intro- 
duction it has distinct advantages, but it raises its own difficulties. 

136 



FULLY CONNECTED SYSTEMS 11/8 

Our present difficulties are, in fact, largely due to this assumption. 
By modifying it we shall not only lessen the difficulties but we 
shall obtain a model more like the real brain. 

The views held about the amount of internal connection in the 
nervous system — its degree of ' wholeness ' — have tended to 
range from one extreme to the other. The ' reflexologists ' from 
Bell onwards recognised that in some of its activities the nervous 
system could be treated as a collection of independent parts. They 
pointed to the fact, for instance, that the pupillary reflex to light 
and the patellar reflex occur in their usual forms whether the 
other reflex is being elicited or not. The coughing reflex follows 
the same pattern whether the subject is standing or sitting. And 
the acquirement of a new conditioned reflex might leave a pre- 
viously established reflex largely unaffected. On the other hand, 
the Gestalt school recognised that many activities of the nervous 
system were characterised by wholeness, so that what happened 
at one point was related to what was happening at other points. 
The two sets of facts were sometimes treated as irreconcilable. 

Yet Sherrington in 1906 had shown by the spinal reflexes 
that the nervous system was neither divided into permanently 
separated parts nor so wholly joined that every event always 
influenced every other. Rather, it showed a richer, and a more 
intricate picture — one in which interactions and independencies 
fluctuated. ' Thus, a weak reflex may be excited from the tail 
of the spinal dog without interference with the stepping-reflex '. 
... ' Two reflexes may be neutral to each other when both are 
weak, but may interfere when either or both are strong '. . . . 
4 But to show that reflexes may be neutral to each other in a 
spinal dog is not evidence that they will be neutral in the animal 
with its whole nervous system intact and unmutilated.' The 
separation into many parts and the union into a single whole are 
simply the two extremes on the scale of ' degree of connected- 
ness '. 

Being chiefly concerned with the origin of adaptation and co- 
ordination, I have tended so far to stress the connectedness of 
the nervous system. Yet it must not be overlooked that adapta- 
tion demands independence as well as interaction. The learner- 
driver of a motor-car, for instance, who can only just keep the 
car in the centre of the road, may find that any attempt at 
changing gear results in the car, apparently, trying to mount 

137 



11/8 DESIGN FOR A BRAIN 

the pavement. Later, when he is more skilled, the act of changing 
gear will have no effect on the direction of the car's travel. Adap- 
tation thus demands not only the integration of related activities 
but the independence of unrelated activities. 

We now, therefore, no longer maintain the restriction of S. 8/6 : 
from now on the main variables may be of any type : full-, part-, 
step-, or null-functions. This freedom makes possible new types 
of ultrastable system, systems still ultrastable and still selective 
for stable fields, but no longer necessarily fully inter-connected 
internally. In particular, if many of the main variables are part- 
functions, the system is able to avoid the earlier-mentioned diffi- 
culty in getting adapted ; it does this by developing partial, 
fluctuating, and temporary independencies within the whole 
without losing its essential wholeness. The study of such systems 
will occupy the remainder of the book. 

References 

Boyd, D. A., and Nie, L. W. Congenital universal indifference to pain. 

Archives of Neurology and Psychiatry, 61, 402 ; 1949. 
Sherrington, C. S. The integrative action of the nervous system. New 

Haven, 1906. 



138 



CHAPTER 12 



Iterated Systems 



12/1. Whereas in the previous chapters we studied a system 
whose main variables were all in intimate connection with one 
another, so that a disturbance applied to any one immediately 
disturbed all the others, we shall now study, for contrast, a system 
composed of the same number of main variables but divided into 
many parts. Each part is assumed to be wholly separated from 
the other parts, and to contain only a few main variables. The 
diagram of immediate effects might appear as in Figure 12/1/1 
which shows, at A, what we have considered in Chapters 8-11, 
and at B what we shall be considering in this chapter. (For 
simplicity, the diagram shows lines instead of arrows.) 






B 



Figure 12/1/1. 

As before, it is assumed that each of the five systems in B 
consists partly of variables belonging to the animal and partly of 
variables belonging to the environment. The relation between 
animal and environment is shown more clearly in Figure 12/1/2. 

Environment. 



f\\ 


1 


// 
7 

// 






\i 



Animal. 
Figure 12/1/2 : Diagrammatic representation of an animal of eight main 
variables interacting with its environment as five independent systems. 

139 



12/2 DESIGN FOR A BRAIN 

Such an arrangement would be shown by any organism that 
reacted to its environment by several independent reactions. In 
such an arrangement each system, still assumed to be ultrastable, 
can change its own step-functions and find its own terminal field 
without effect on what is happening in the others. We shall say 
that the whole consists of iterated ultrastable systems. 

Since each system is ultrastable it can adapt and learn inde- 
pendently of the others. That such independent, localised learn- 
ing can occur within one animal was shown by Parker in the 
following experiment : 

4 If a sea-anemone is fed from one side of its mouth, it will 
take in, by means of the tentacles on that side, one fragment 
of food after another. If now bits of food be alternated with 
bits of filter paper soaked in meat juice, the two materials 
will be accepted indiscriminately for some eight or ten trials, 
after which only the meat will be taken and the filter paper 
will be discharged into the sea water without being brought 
to the mouth. If, after having developed this state of affairs 
on one side of the mouth, the experiment is now transferred 
to the opposite side, both the filter paper and the meat will 
again be taken in till this side has also been brought to a state 
of discriminating.' 

12/2. If we start a set of iterated ultrastable systems, and 
observe the set's behaviour, noting particularly at each moment 
how many of the systems have arrived at a terminal field, we 
shall find that the set, regarded as a whole, shows the following 
properties. 

The proportion which is adapted is no longer restricted to the 
two values ' all ' or ' none '. In fact, if the systems are many, 
the degrees of adaptation which the whole can show will be as 
many. A whole which consists of iterated systems will therefore 
show in its adaptation a gradation which was seen (S. 11/2) to 
be lacking in the fully-connected ultrastable system. 

A second property is that when one system has arrived at a 
terminal field, the changes of the other systems will not cause the 
loss of the first field. In other words, while the later adaptations 
are being found, the earlier are conserved. A whole which con- 
sists of iterated systems will therefore show some conservation of 
adaptations. 

A third property is that, as time passes, the number of systems 

140 



ITERATED SYSTEMS 12/3 

which are adapted will increase, or may stay constant, but cannot 
decrease (in the conditions assumed here : more complex conditions 
are discussed later). If the number of stable systems is regarded 
as measuring, in a sense, the degree of adaptation achieved by the 
whole, then, in a whole which consists of iterated systems, the 
degree of adaptation tends always to increase. The whole will 
therefore show a progression in adaptation. 

12/3. Let us now compare the two types of system, (a) the 
fully connected, and (b) the iterated, in the times they take, 
on the average, to reach terminal fields, other things, including 
the number of main variables, being equal. (The calculation 
can only be approximate but the general conclusion is unam- 
biguous.) 

We start with a system of N main variables and want to find, 
approximately, how long the system will take on the average to 
reach the condition where all N main variables belong to systems 
with stable fields , Three arrangements will be examined ; they are 
extreme in type, but they illustrate the possibilities. (1) All the 
N main variables belong to one system, so that to stabilise all 
N a field must stabilise all simultaneously. (2) Each main vari- 
able is in a system which includes it alone, and where the systems 
are related in such a way that only after the first is stabilised can 
the second start to get stabilised, and so on in succession. (3) 
Each system, also containing only one main variable, proceeds 
independently to find its own stability. 

In order to calculate how long the three types will take, suppose 
for simplicity that each main variable has a constant and inde- 
pendent probability p of becoming stable in each second. 

The type in which stability can occur only when all the N 
events are favourable simultaneously will have to wait on the 

average for a time given by 1\ = —. The type in which sta- 
bility can occur only by the variables achieving stability in succes- 
sion will have to wait on the average for a time given by 1\ = N/p. 
And the type in which the variables proceed independently to 
stability will have to wait on the average for a time which is 
difficult to specify but which will be of the order of T 3 = 1/p. 
These three estimates of the time taken are of interest, not for 
their quantitative exactness, but for the fact that they tend to 

141 



12/4 DESIGN FOR A BRAIN 

have widely different values. Some numerical values will be 

calculated in order to demonstrate the differences. The values 

have not been specially selected, and if the reader will substitute 

some values of his own he will probably find that his values lead 

to essentially the same conclusions as are reached here. 

Suppose that the chance of any one variable becoming stable 

in a given second is a half. If we are testing a system with a 

thousand variables, then N = 1000 

and J\ = 2 1000 sees. 

rr 100 ° 

1 2 = — £- sees. 

T 3 = about \ sec. 
When these are converted to more ordinary numbers, we find that 
the three quantities differ widely. T z is about a half-second, T 2 
is about 8 minutes, and 1\ is about 3 X 10 291 centuries. The 
last number, if written in full, would consist of a 3 followed by 
about five lines of zeros. 1\ and T 2 are moderate, but T 1 is so 
vast as to be outside even astronomical duration. 

This example is typical. What it means in general is that 
when N is large, it is not possible to get stability if all N must 
find some favourable feature simultaneously. The calculation 
confirms the statement of S. 11/7 that it is not reasonable to 
assume that 10 10 neurons have formed a stable field by waiting 
for the fortuitous occurrence of one field which stabilises all. 

12/4. The argument may also be viewed from a different angle. 
When the system of a thousand variables could achieve stability 
only by the occurrence of a field which was favourable to all at 
once, it had to wait, on the average, through 3 X 10 291 centuries. 
But if its conditions were changed so that the variables could 
become stable in succession or independently, then the time taken 
dropped to a few minutes or less. In other words, what was, for 
all practical purposes, an impossibility under the first condition 
became, under the second and third conditions, a ready possibility. 
It is difficult to find a real example which shows in one system 
the three ways of progression to stability, for few systems are 
constructed so flexibly. It is, however, possible to construct, by 
the theory of probability, examples which show the differences 
referred to. Thus suppose that, as the traffic passes, we note the 
final digit on each car's number-plate, and decide that we want 

142 



ITERATED SYSTEMS 12/5 

to see cars go past with the final digits 0,1, 2, 3, 4, 5, 6, 7, 8, 9, in 
that order. If we insist that the ten cars shall pass consecutively, 
then on the average we shall have to wait till about 10,000,000,000 
cars have passed : for practical purposes such an event is impos- 
sible. But if we allow success to be achieved by first finding a 
1 ', then finding a 4 1 ', and so on until a l 9 ' is seen, then the 
number of cars which must pass will be about fifty, and this 
number makes 4 success ' easily achievable. 

12/5. A well-known physical example illustrating the difference 
is the crystallisation of a solid from solution. When in solution, 
the molecules of the solute move at random so that in any given 
interval of time there is a definite probability that a given molecule 
will possess a motion and position suitable for its adherence to 
the crystal. Now the smallest visible crystal contains billions of 
molecules : if a visible crystal could form only when all its mole- 
cules happened simultaneously to be properly related in position 
and motion to one another, then crystallisation could never occur : 
it would be too improbable. But in fact crystallisation can occur 
by succession, for once a crystal has begun to form, a single 
molecule which happens to possess the right position and motion 
can join the crystal regardless of the positions and motions of the 
other molecules in the solution. So the crystallisation can pro- 
ceed by stages, and the time taken resembles T 2 rather than 2\. 
We may draw, then, the following conclusion. A compound 
event that is impossible if the components have to occur simul- 
taneously may be readily achievable if they can occur in sequence 
or independently. 

Reference 

Parker, G. The evolution of man. New Haven, 1922. 



143 



CHAPTER 13 

Disturbed Systems and Habituation 



13/1. We have seen that ultrastable systems are subject to two 
conflicting requirements : complexity and speed. The system 
with abundant internal connections, though able to represent 
a complex and well-integrated organism and environment, 
requires, at least in the form so far studied, almost unlimited time 
for its adaptation. On the other hand, the same number of main 
variables, divided into many independent parts, achieves adapta- 
tion quickly, but cannot represent a complex biological system. 
There are, however, intermediate forms that can combine, to 
some extent, the advantages of these two extremes. Since the 
properties of the intermediate forms are somewhat subtle we 
shall have to proceed by small steps. As a first step I shall 
examine in this chapter the properties of ultrastable systems that 
are no longer completely isolated, as has been assumed so far, but 
are subject to some slight disturbance from the outside. 

13/2. Before entering the subject, I must make clear a point of 
method that will be used frequently. In Chapter 8, the discussion 
of the ultrastable system necessarily paid so much attention to the 
process by which the terminal field was reached that some loss of 
proportion occurred; for the focusing of attention suggested 
that the system spent most of its time reaching a terminal field, 
whereas in the living organism this process may occupy only a few 
moments — a time unimportant in comparison with the remainder 
of the organism's life during which the terminal field will act 
repeatedly to keep the essential variables within limits. 

From this point of view the terminal field is more important 
than the preceding fields simply because it is permanent while 
the others are transient. As we increase the time over which the 
system is observed, so do the transient fields become negligible. 
The same principle is used in the Darwinian theory of natural 

144 



DISTURBED SYSTEMS AND HABITUATION 13/4 

selection where, although it is recognised that mutations and re- 
combinations of defective viability can occur, yet as the processes 
of evolution are viewed over an increasing range of time, so 
do these defectively adapted individuals sink into insignificance. 
Statistical mechanics, too, uses the same principle, for it excludes 
an event by proving that its occurrence is not impossible but 
infrequent. Sometimes we shall not be able to distinguish even 
the transient from the permanent, but only the lesser from the 
greater persistence. Nevertheless, the distinction may be im- 
portant, especially if the small difference acts repeatedly and 
cumulatively ; for what is feeble on a single action may be over- 
whelming on incessant repetition. 

It will be suggested later (S. 16/6) that the animal's behaviour 
depends not on one system but on many, so that what counts is 
not the peculiarity of one particular field but the average proper- 
ties of many. In the discussion we shall therefore notice the 
average properties and the tendencies rather than the individual 
peculiarities of the various fields. 

Effects of small random disturbances 

13/3. A disturbance may affect variables or parameters. If it 
affects the variables, the system will undergo a sudden change of 
state ; in the phase-space the representative point would be 
displaced suddenly from one line of behaviour to another. If 
it affects a parameter, there will occur a sudden change of field : 
the representative point will be affected only mediately. 

13/4. We shall now examine the effect on an ultrastable system 
of small, occasional, and random disturbances applied to the 
variables. I assume at first that the displacements are distributed 
in all directions in the phase-space. 

A displacement may make the representative point meet a 
critical state it would not otherwise have met ; then the dis- 
placement destroys the field. The three fields of Figure 13/4/1 
show some of the consequences. In fields A and C the undis- 
turbed representative points will go to, and remain at, the resting 
states. When they are there, a leftwards displacement sufficient 
to cause the representative point of A to encounter the critical 
states may be insufficient if applied to C ; so Cs field may survive 

145 



13/4 DESIGN FOR A BRAIN 

a displacement that destroyed A's. Similarly a displacement 
applied to the representative point on the resting cycle in B is 
more likely to change the field than if applied to C. A field like 
C, therefore, with its resting states compact and near the centre 
of the region, tends to have a higher immunity to displacement 
than fields whose resting states or cycles go near the edge of the 



Figure 13/4/1 : Three fields of an ultrastable system, differing in their 
liability to change when the system is subjected to small random dis- 
turbances. (The critical states are shown by the dots.) 

region. (A quantitative discussion of the tendency is given in 
S. 23/4.) 

If the disturbances fall on a large number of iterated ultrastable 
systems, the probabilities become actual frequencies. We can 
then predict that if iterated ultrastable systems are subjected to 
repeated small occasional and random disturbances, the average 
terminal field will tend to the form C. 

13/5. How would this tendency show itself in the behaviour of 
the living organism ? 

In S. 8/7 we noticed that a field may be terminal and yet 
show all sorts of bizarre features : cycles, resting states near the 
edge of the region, stable and unstable lines mixed, multiple 
resting states, multiple resting cycles, and so on. These possi- 
bilities obscured the relation between a field's being terminal and 
its being suitable for keeping essential variables within normal 
limits. But a detailed study was not necessary ; for we have 
just seen that all such bizarre fields tend selectively to be destroyed 
when the system is subjected to small, occasional, and random dis- 
turbances. Since such disturbances are inseparable from practical 
existence, the process of ' roughing it ' tends to cause their replace- 

146 



DISTURBED SYSTEMS AND HABITUATION 13/7 

ment by fields of ' normal ' stability (S. 20/2) that look like C 
of Figure 13/4/1 and act simply to keep the representative point 
well away from the critical states. 



Effects of repeated stimuli 

13/6. So far we have studied only the effects of irregular dis- 
turbances : what of the regular ? By the argument of S. 6/6, all 
such can be considered as * stimuli ' and are of two types : a 
sudden change of parameter-value, and a sudden jump of the 
representative point. The two types will be considered separately. 

The effect on an ultrastable system of an alternation of a 
parameter between two values has already been described in S. 
11/4, where Figure 11/4/1 showed how the ultrastable system is 
automatically selective for any set of step-function values which 
gives stability with both the parameter- values. 

The facts can also be seen from another point of view. If we 
start alternating the parameter and observe the response of the step- 
functions we shall find that at first they change, and that after a 
time they stop changing. The responses, in other words, diminish. 

13/7. Next consider the effect of repetitions of the other type of 
stimulus — the displacement of 
the representative point. Its 
effect can readily be found by 

asking what sort of field can be • ' 

terminal. Suppose, for instance, ' 

that the displacement was a A ' Q 

movement to the left through » 

the distance shown by the arrow 
below Figure 1 3/7/1 . It is easy 
to see that a field, to be terminal 
in spite of this displacement, 
must have its resting state 
within B. If the constant dis- 
placement is applied from time *"* • 
to time to an ultrastable system FlGU Rf 13/7/1 : Region of an ultra- 
, r. i t , . stable system. The representative 
whose fields have resting states point must stay in B if the field is 
distributed over both A and B, to b f imm " n e to a displacement 
., • i /> i i • , . equal and parallel to that shown 
then terminal fields with resting by the arrow. 

147 






13/8 DESIGN FOR A BRAIN 

state in A will be destroyed ; but the first with resting state in B 
will be retained. The displacement will then stop causing step- 
function changes. So if we regard the application of the constant 
displacement as • stimulus ', and the step-function and main- 
variable changes as ' response ', then we shall find that the 
response to the stimulus tends to diminish. 

13/8. This particular process cannot be shown on the homeostat, 
for its resting state is always at the centre, but it will demon- 
strate a related fact. If two fields (Figure 13/8/1) each have a 



B 




Figure 13/8/1. 

resting state at the centre and the line of one (A) from a constant 
displacement returns by a long loop meeting critical states while 
the return path of the other (B) is more direct, then the applica- 
tion of the displacement will destroy A but not B. In other 
words, a set of step-function values which gives a large ampli- 
tude of main-variable movement after a constant displacement 
is more likely to be replaced than a set which gives only a small 
amplitude. 

The process is shown in Figure 13/8/2. Two units were joined 
1 — > 2. The effect of 1 on 2 was determined by 2's uniselector, 
which changed position if 2 exceeded its critical states. The 
operator then repeatedly disturbed 2 by moving 1, at D. As 
often as the uniselector transmitted a large effect to 2, so often 
did 2 shift its uniselector. But as soon as the uniselector arrived 
at a position that gave a transmission insufficient to bring 2 to its 
critical states, that position was retained. So under constant 
stimulation by D the amplitude of 2's response tended to diminish. 

The same process in a more complex form is shown in Figure 

148 



DISTURBED SYSTEMS AND HABITUATION 



13/9 



13/8/3. Two units are interacting : 1 ^± 2. Both effects go 
through the uniselectors, so the whole is ultrastable. At each D, 
the operator displaced l's magnet through a constant distance. 
On the first c stimulation ', 2's response brought the system 
to its critical states, so the ultrastability found a new terminal 



D 




Time 



Figure 13/8/2 : Homeostat tracing. At each D, l's magnet is displaced 
by the operator through a fixed angle. 2 receives this action through 
its uniselector. When the uniselector's value makes 2's magnet meet 
the critical states (shown dotted) the value is changed. After the 
fourth change the value causes only a small movement of 2, so the value 
is retained permanently. 



-\rv 



Time 



Figure 13/8/3 : Homeostat arranged as ultrastable system with two units 
interacting. At each D the operator moved l's magnet through a fixed 
angle. The first field such that D does not cause a critical state to be 
met is retained permanently. 

field. The second stimulation again evoked the process. But 
the new terminal field was such that the displacement D no 
longer caused 2 to reach its critical states ; so this field was 
retained. Again under constant stimulation the response had 
diminished. 



13/9. For completeness we will now consider the effects of these 
disturbances in combination. The combination of repeated 
constant displacements with small random disturbances yields 
little of interest. But the combination of an alternation of para- 
meter-value with small random disturbances is worth notice. 

149 l 



13/9 



DESIGN FOR A BRAIN 



From S. 11/4 it is known that the alternation of a parameter 
p between two values, p' and p", will result in the emergence of 
two stable fields. We might get a pair like A and B of Figure 
13/9/1. As the parameter p alternates between p' and p", so 





Figure 13/9/1 : Two possible terminal fields : A, when p has the value p' 
and B, when it has the value p" . (Critical states shown as dots.) 




Figure 13/9/2 : When the parameter is constant at p', the representative 
point will follow the path from )3 to a ; when at p" the point follows 
the path from a to j8. 

will the field of the system's main variables alternate between 
A and B. If p alternates slowly in comparison with the move- 
ment of the representative point, the point will follow the circuit 
of Figure 13/9/2, going from a to /? when p is changed from 
p' to p'\ and returning to a when p is returned to p' . 

Suppose now that small random disturbances are applied to 

150 



DISTURBED SYSTEMS AND HABITUATION 13/10 

two such systems (C and D) with circuits such as arc shown in 
Figure 13/9/3. We can predict (by S. 13/4) that a system of 
type D, with a short and central circuit, will have a higher 
immunity to random disturbance than a system of type C. 
Maximal immunity will be shown by systems in which a and ft 
coalesce at the centre of the region. 




<& 



D 



Figure 13/9/3. 

If there are many systems like C and D, the probabilities become 
actual frequencies. As the less resistant fields are destroyed while 
the more resistant remain, the average movement of the repre- 
sentative point, as the parameter is alternated between p' and 
p", will change from a large circuit like C towards a small central 
circuit like D. So both the number of step-function changes and 
the range of movement evoked by the stimulus p will diminish. 

Habituation 

13/10. Some uniformity is now discernible in the responses of an 
ultrastable system to repeated stimuli. There is a tendency for 
the response, whether measured by the number of step-functions 
changing or by the range of movement of the main variables, 
to diminish. The diminution is not due to any triviality of 
definition or to any peculiarity of the homeostat : it follows from 
the basic fact, inseparable from any delicate or ultrastable system, 
that large responses tend, if there is feedback, to destroy the 

151 



13/11 DESIGN FOR A BRAIN 

conditions that made them large, while small responses do not 
destroy the conditions that made them small. 

13/11. In animal behaviour the phenomenon of ' habituation ' 
is met with frequently : if an animal is subjected to repeated 
stimuli, the response evoked tends to diminish. The change has 
been considered by some to be the simplest form of learning. 
Neuronic mechanisms are not necessary, for the Protozoa show it 
clearly : 

4 Amoebae react negatively to tap water or to water from a 
foreign culture, but after transference to such water they 
behave normally.' 

4 If Paramecium is dropped into J% sodium chloride it 
at once gives the avoiding reaction ... If the stimulating 
agent is not so powerful as to be directly destructive, the 
reaction ceases after a time, and the Paramecia swim about 
within the solution as they did before in water.' (Jennings.) 

Fatigue has sometimes been suggested as the cause of the 
phenomenon, but in Humphrey's experiments it could be excluded. 
He worked with the snail, and used the fact that if its support is 
tapped the snail withdraws into its shell. If the taps are repeated 
at short intervals the snail no longer reacts. He found that when 
the taps were light, habituation appeared early ; but when they 
were heavy, it was postponed indefinitely. This is the opposite 
of what would be expected from fatigue, which should follow more 
rapidly when the heavier taps caused more vigorous withdrawals. 

The nature of habituation has been obscure, and no explanation 
has yet received general approval. The results of this chapter 
suggest that it is simply a consequence of the organism's ultra- 
stability, a by-product of its method of adaptation. 



Reference 
Humphrey, G. The nature of learning. London, 1933. 



152 



CHAPTER 14 



Constancy and Independence 



14/1. Several times we have used, without definition, the con- 
cept of one variable or system being ' independent ' of another. 
It was stated that a system, to be absolute, must be 4 properly 
isolated ' ; some parameters in S. 6/2 were described as 4 ineffec- 
tive ' ; and iterated ultrastable systems were defined as ' wholly 
independent ' of each other. So far a simple understanding has 
been adequate. But as it is now intended to treat of systems 
that are neither wholly joined nor wholly separated, a more 
rigorous method is necessary. 

The concept of the l independence ' of two dynamic systems 
might at first seem simple : is not a lack of material connection 
sufficient ? Examples soon show that this criterion is unreliable. 
Two electrical parts may be in firm mechanical union, yet if 
the bond is an insulator the two parts may be functionally inde- 
pendent. And two reflex mechanisms in the spinal cord may be 
inextricably interwoven, and yet be functionally independent. 

On the other hand, one system may have no material con- 
nection with another and yet be affected by it markedly : the 
radio receiver, for instance, in its relation to the transmitter. 
Even the widest separation we can conceive — the distance between 
our planet and the most distant nebulae — is no guarantee of 
functional separation ; for the light emitted by those nebulae 
is yet capable of stirring the astronomers of this planet into con- 
troversy. The criterion of connection or separation is thus useless. 

14/2. This attempted criterion obtained its data by a direct 
examination of the real ' machine '. The examination not only 
failed in its object but violated the rule of S. 2/8. What we 
need is a test that uses only information obtained by primary 
operations. 

It is convenient to approach the subject by first clarifying 
what we mean by one system c controlling ' or ' affecting ' another. 

153 



14/2 



DESIGN FOR A BRAIN 



Our understanding has been greatly increased by the development 
during recent years of the science of 4 cybernetics '. The word — 
from KvfiepvrjTf]£, a steersman — was coined by Professor Wiener to 
describe the science which, though really dating back to Watt 
and his governor for steam-engines, has developed partly as a 
result of the extraordinary properties of the thermionic vacuum 
tube, and partly as a result of the urgent demands during the 
last war for complex calculating and controlling machinery such 
as predictors for gun- and bomb-sights, automatically controlled 
searchlights, automatically controlled anti-aircraft guns, and 
electronic computors. 

When a radar-installation passes information about the posi- 
tion of an aeroplane to a predictor, and the predictor emits 
instructions which determine, either manually or automatically, 
the laying of an anti-aircraft gun, we can write simply enough : — 



Radar 
set 



Predictor 



> Gun 



but what do the arrows mean ? what is transmitted from box 
to box ? Energy ? No, says cybernetics — information. 

If we turn to simple machines for guidance, we will probably 
be misled. When my finger strikes the key of a typewriter, 
the movement of my finger determines the movement of the 
type ; and the finger also supplies the energy necessary for the 
type's movement. The diagram 



B 



would state, in this case, both that energy, measurable in ergs, 
is transmitted from A to B, and also that the behaviour of B 
is determined by, or predictable from, that of A. If, however, 
power is freely available to B, the transmission of energy from 
A to B becomes irrelevant to the question of the control exerted. 
It is easy, in fact, to devise a mechanism in which the flow of 
both energy and matter is from B to A and yet the control is 
exerted by A over B. Thus, suppose B contains a compressor 
which pumps air at a constant rate into a cylinder creating a 
pressure that is shown on a dial. From the cylinder a pipe goes 
to A, where there is a tap which can allow air to escape and 

154 



CONSTANCY AND INDEPENDENCE 



14/3 



can cause the pressure in the cylinder to fall. Now suppose a 
stranger comes along ; he knows nothing of the internal mech- 
anism, but tests the relations between the two variables : A, 
the position of the tap, and B, the reading on the dial. By 
direct testing he soon finds that A controls Z?, but that B has 
no effect on A. The direction of control has thus no necessary 
relation to the direction of flow of either energy or matter when 
the system is such that all parts are supplied freely with energy. 

14/3. The factual content of the concept of one variable c con- 
trolling ' another is now clear. A ' controls ' B if B's behaviour 
depends on A, while A's does not depend on B. But first we 
need a definition of ' independence '. Given a system that includes 
two variables A and B, and two lines of behaviour ivhose initial 
states differ only in the values of B, A is independent of B if A's 
behaviours on the tzvo lines are identical. The definition can be 
illustrated on the data in Table 14/3/1. On the two lines of 













Time 








Variable 































1 


2 


3 


4 


1 




A 


12 


19 


33 


49 


55 




1 


B 


40 


39 


31 


21 


14 


Line 




C 


18 


22 


28 


37 


47 




A 


12 


19 


33 


49 


55 




2 


B 


25 


18 


12 


7 


4 






C 


18 


21 


20 


17 


5 



Table 14/3/1 : Two lines of behaviour of a three- variable absolute system. 



behaviour the initial states are equal except for the values of B. 
The subsequent behaviours of A on the two lines are identical. 
So A is independent of B. (Independence within the range 
covered by the table in no way restricts what may happen 
outside it.) By the definition, C is not independent of B. 

By ' dependent ' will be meant simply 4 not independent '. 

The definition is given primarily by reference to two lines of 
behaviour, for only in this form is the result of the criterion 

155 



14/4 DESIGN FOR A BRAIN 

always unambiguous. Other criteria might be confused by some 
of the fields that ingenuity can construct. But often a simple 
uniformity holds. Two variables may be independent over all 
such pairs of initial states ; and sometimes all variables of one 
set may be independent of all variables of another set : a system 
R is independent of a system S if every variable in It is independent 
of every variable in S, all possible pairs being considered. Some 
region of the field is understood to be given before the test is 
applied. 

14/4. To illustrate the definition's use, and to show that its 
answers accord with common experience, here are some examples. 

If a bacteriologist wishes to test whether the growth of a 
micro-organism is affected by a chemical substance, he prepares 
two tubes of nutrient medium containing the chemical in different 
concentrations but with all other constituents equal ; he seeds 
them with equal numbers of organisms ; and he observes how 
the increasing numbers of organisms compare in the two tubes 
from hour to hour. Thus he is observing the numbers of organisms 
after two initial states that differed only in the concentrations 
of chemical. 

To test whether an absolute system is dependent on a para- 
meter, i.e. to test whether the parameter is ' effective ', we observe 
the system's behaviour on two occasions when the parameter 
has different values. Thus, to test whether a thermostat is 
really affected by its regulator one sets the regulator at some 
value, checks that the temperature is at its usual value, and 
records the subsequent behaviour of the temperature ; then one 
returns the temperature to its previous value, changes the posi- 
tion of the regulator, and observes again. A change of behaviour 
implies an effective regulator. (Here we have used the fact that 
by S. 21/4 we can take a null-function into the system without 
altering its absoluteness, for the change is only formal.) 

Finally, an example from animal behaviour. Parker tested 
the sea-anemone to see whether the behaviour of a tentacle was 
independent of its connection with the body. 

* When small fragments of meat are placed on the tentacles 
of a sea-anemone, these organs wind around the bits of 
food and, by bending in the appropriate direction, deliver 
them to the mouth.' 

156 



CONSTANCY AND INDEPENDENCE 14/6 

(He has established that the behaviour is regular, and that the 
system of tentacle-position and food-position is approximately 
absolute. He has described the line of behaviour following the 
initial state : tentacle extended, food on tentacle.) 

4 If, now, a distending tentacle on a quiet and expanded 
sea-anemone is suddenly seized at its base by forceps, cut 
off and held in position so that its original relations to the 
animal as a whole can be kept clearly in mind, the tentacle 
will still be found to respond to food brought in contact 
with it and will eventually turn toward that side which was 
originally toward the mouth.' 

(He has now described the line of behaviour that follows an initial 
state identical with the first except that the null-function ' con- 
nection with the body ' has a different value. He observed that 
the two behaviours of the variable ' tentacle-position ' are identi- 
cal.) He draws the deduction that the tentacle-system is, in this 
aspect, independent of the body-system : 

4 Thus the tentacle has within itself a complete neuro- 
muscular mechanism for its own responses.' 

The definition, then, agrees with what is usually accepted. 
Though clumsy in simple cases, it has the advantage in complex 
cases of providing a clear and precise foundation. By its use 
the independencies within a system can be proved by primary 
operations only. 

14/5. In an absolute system it is not generally possible to 
assign the dependencies and independencies arbitrarily. For if 
x is dependent on y, and y is dependent on z, then x must neces- 
sarily be dependent on z. This is evident, for when s's initial 
state is changed, i/'s behaviour is changed ; and these changed 
values of y, acting in an absolute system, will cause x , s behaviour 
to change. So the observer will find that a change in z 9 s initial 
state is followed by a change in as's behaviour. (A formal proof 
is given in S. 24/11.) 

14/6. We can now see that the method for testing an imme- 
diate effect, described in S. 4/12, is simply a test for independence 
applied when all the variables but two are held constant. The 
relation can be illustrated by an example. Suppose three real 
machines are linked so that their diagram of immediate effects is 

% — > y — > x. 
157 



14/6 



DESIGN FOR A BRAIN 



The system's responses to tests for independence will show that 
y is independent of x, and that z is independent of both. The 
same set of independencies would be found if we tested the three 
machines when their linkages were 



The distinction appears when we test the immediate effect be- 
tween z and x. For if in both cases we fix y, we shall find in 
the first that x is independent of 2, but in the second that x is 
not independent of z. 

Given a system's diagram of immediate effects, its diagram 
of ultimate effects is formed by adding to every pair of arrows 
joined tail to head a third arrow going from tail to head, like 
2 — > x above, and by repeating this process until no further 
additions are possible. Thus, the diagram of immediate effects 
I in Figure 14/6/1 would yield the diagram of ultimate effects II. 




>■ 2 



K2 




n 



Figure 14/6/1. 



The diagram of ultimate effects shows directly and completely 
the independencies in the system. Thus, from II of the figure 
we see that variable 1 is independent of 2, 3, and 4, and that 
the latter three are dependent on all the others. 

14/7. If, in a system, some of the variables are independent 
of the remainder, while the remainder are not independent of 
the first set, then the first set dominates the remainder. Thus, 
in Figure 14/6/1, variable 1 dominates 2, 3, and 4. And in 
the diagram of S. 6/6 the animal dominates the recorders. 



The effects of constancy 

14/8. So far the independencies have been assumed permanent : 
we now study the conditions under which they can alter. 
Suppose an absolute system of eight variables has the diagram 

158 



CONSTANCY AND INDEPENDENCE 



14/9 



of immediate effects shown in Figure 14/8/1. What properties 
must the three variables B have if the systems A and C are to 
become independent and absolute ? The question has not only 
theoretical but practical importance. Many experiments require 
that one system be shielded from effects coming from others. 
Thus, a system using magnets may have to be shielded from the 
effects of the earth's magnetism ; or a thermal system may have 
to be shielded from the effects of changes in the atmospheric tem- 




Figure 14/8/1. 

perature ; or the pressure which drives blood through the kidneys 
may have to be kept independent of changes in the pulse-rate. 
A first suggestion might be that the three variables B should 
be removed. But this conceptual removal corresponds to no 
physical reality : the earth's magnetic field, the atmospheric 
temperature, the pulse-rate cannot be ' removed '. In fact the 
answer is capable of proof (S. 24/15) : that A and C should he 
independent and absolute it is necessary and sufficient tlmt the 
variables B should be null-functions. In other words, A and C 
must be separated by a wall of constancies. 



14/9. Here are some illustrations to show that the theorem 
accords with common experience. 

(a) If A (of Figure 14/8/1) is a system in which heat-changes 
are being studied, B the temperatures of the parts of the con- 
tainer, and C the temperatures of the surroundings, then for A 
to be isolated from C and absolute, it is necessary and sufficient 
for the B's to be kept constant, (b) Two electrical systems joined 
by an insulator are independent, if varying slowly, because 
electrically the insulator is unvarying, (c) The centres in the 
spinal cord are often made independent of the activities in the 
brain by a transection of the cord ; but a break in physical con- 

159 



14/10 



DESIGN FOR A BRAIN 



tinuity is not necessary : a segment may be poisoned, or anaes- 
thetised, or frozen ; what is necessary is that the segment should 
be unvarying. 

Physical separation, already noticed to give no certain inde- 
pendence, is sometimes effective because it sometimes creates an 
intervening region of constancy. 

14/10. The example of Figure 14/8/1 showed one way in which 
the constancy of a set of variables could affect the independencies 
within a system. The range of ways is, however, much greater. 
To demonstrate the variety we need a rule by which we can 
make the appropriate modifications in the diagram of ultimate 
effects when one or more of the variables are held constant. The 
rule is proved in S. 24/14 : — Take the diagram of immediate 
effects. If a variable V is constant, remove all arrows whose 
heads are at V ; then, treating this modified diagram as one of 
immediate effects, complete the diagram of ultimate effects, using 

1-^2 



\/\ 



4-* 



R C D F 

1^=§=±=2 \-±-*z2 1r< 2 1 t 2 




[IXIIXH/I 



^3 4 ± — ^ 3 4-« 3 4-* 



Figure 14/10/1 : If a four-variable system has the diagram of immediate 
effects A, and if 1 and 2 are part-functions, then its diagram of ultimate 
effects will be B, C, D or E as none, 1, 2, or both 1 and 2 become 
inactive, respectively. 

the rule of S. 14/6. The resulting diagram will be that of the 
ultimate effects, and therefore of the independencies, when V is 
constant. (It will be noticed that the effect of making V constant 
cannot be deduced from the diagram of ultimate effects alone.) 
Thus, if the system of Figure 14/10/1 has the diagram of imme- 
diate effects A, then the diagram of ultimate effects will be B, C, 
D or E according as none, 1, 2, or both 1 and 2 are constant, 
respectively. 

It can be seen that with only four variables, and with only 
two of the four possibly becoming constant, the patterns of 

160 



CONSTANCY AND INDEPENDENCE 14/11 

independence show a remarkable variety. Thus, in C, 1 domi- 
nates 3 ; but in D, 3 dominates 1. As the variables become 
more numerous so does the variety increase rapidly. 

The multiplicity of inter-connections possible in a telephone 
exchange is due primarily to the widespread use of temporary 
constancies. The example serves to remind us that 8 switching ' 
is merely one of the changes producible by a re-distribution of 



+*• 



**• 



B 

Figure 14/10/2. 

constancies. For suppose a system has the diagram of imme- 
diate effects shown in Figure 14/10/2. If an effect coming from 
C goes down the branch AD only, then, for the branch BE to 
be independent, B must be constant. How the constancy is 
obtained is here irrelevant. When the effect from C is to be 
4 switched ' to the BE branch, B must be freed and A must 
become constant. Any system with a ' switching ' process must 
use, therefore, an alterable distribution of constancies. Con- 
versely, a system whose variables can be sometimes fluctuating and 
sometimes constant is adequately equipped for switching. 

14/11. At this point it is convenient to consider what degree 
of independence is shown in a system if some part is not directly 
affected by some other part. To take an extreme case, to what 
extent are two parts joined functionally if they have only a 



W x \ 



A B 

Figure 14/11/1. 

single variable in common — the parts A and B in Figure 14/11/1, 
for instance, which share only the variable x ? It is shown in 
S. 24/17 that if x is a full-function capable of unrestricted varia- 
tion, then the two parts A and B are as effectively joined as if 

161 



14/12 DESIGN FOR A BRAIN 

they had many more direct effects bridging the gap. Construc- 
tion of the diagram of ultimate effects provides a simple proof. 
The explanation is that each system affects, and is affected by, 
not only aj's value but also a?'s first, second, and higher derivatives 
with respect to time. These act to provide a richness of func- 
tional connection that is not evident at first glance. 

Part- functions 

14/12. In S. 14/8 we saw that if a whole system is to be divided 
into independent parts some intervening variables must become 
constant. It follows that if the independence is to be tem- 
porary, being sometimes present and sometimes absent, the inter- 
vening variables must be sometimes constant and sometimes 
varying : they must, in short, be part-functions. This class of 
variable will therefore now be considered. 

A part-function was defined in S. 7/1 as a variable which, 
over some interval of observation, was constant over some finite 
intervals and fluctuated over some finite intervals. It is not 
implied that the constant values are all equal. The definition 
refers solely to the variable's observed behaviour, making no 
reference to any cause for such behaviour ; though there will 
usually be some definite physical reason to account for this way 
of behaving. A part-function will be said to be ' active ', or 
4 inactive ', at a given moment according to whether it is, or 
is not, varying. As the amount of time spent active tends 
to 'all', or 'none', so does the part-function tend to full-, or 
step-, function form. The part-function thus fills the gap be- 
tween the two types, and may be expected to have intermediate 
properties. 

14/13. Here are some examples. Like the step-function, it is 
met with much more commonly in the real world than in books. 
— The pressure on the brake-pedal during a car journey. The 
current flowing through a telephone during the day. The posi- 
tion co-ordinates of an animal, such as a frog or grasshopper, 
that moves intermittently. The pressure on the sole of the 
foot during walking. The activity of pain receptors, if they are 
activated only intermittently. The rate of secretion of saliva in 
an experiment on the conditioned reflex. The rate at which 

162 



CONSTANCY AND INDEPENDENCE 14/15 

water is being swallowed (ml. /sec.) by a land animal observed 
over several days. The sexual activities of a stag during the 
twelve months. The activity in the mechanisms responsible for 
reflexes which act only intermittently : vomiting, sneezing, 
shivering. 

14/14. The property of ' threshold ' leads often to behaviour of 
part-function form. For if x dominates y, and if, when x is less 
than some value, y remains constant, while if, when x is greater 
than the value, y fluctuates as some function of x, then, if x is 
a full-function and fluctuates across the threshold, y will behave 
as a part-function. In the nervous system, and in living matter 
generally, threshold properties are widespread ; part-functions 
may therefore be expected to be equally widespread. 

Systems containing part-functions 

14/15. Having earlier examined the properties of systems con- 
taining null-functions (S. 7/7), and step-functions (S. 7/8 et seq.), 
we will now examine the properties of systems containing part- 
functions. It is convenient to suppose at first that the system 
is composed of them exclusively. 

Even when not at a resting state, some of such a system's 
variables may be constant. If the system is composed of part- 
functions which are active for most of the time, the system will 
show little difference in behaviour from one composed wholly of 
full-functions. But if the part-functions are active only at in- 
frequent intervals then, as the system traverses some line of 
behaviour, inspection will show that only some of the variables 
are changing, the remainder being constant. Further, if observed 
on two lines of behaviour, the set of variables which were active on 
the first line will in general be not the same as the set active 
on the second. That this may be so can be seen by considering 
its field. 

The field of an absolute system which contains part-functions 
has the peculiarity that the lines of behaviour often run in a 
sub-space. Thus, over an interval when all the variables but 
one are inactive, the line will run in a straight line parallel to 
the axis of the active variable. If all but two are inactive, the 
line will run in a plane parallel to that which contains the axes 

163 



14/15 



DESIGN FOR A BRAIN 



of the two active variables ; and so on. If all the variables 
are inactive, the line becomes a point. Thus a three-variable 

system might give the line of behaviour 
shown in Figure 14/15/1. 

An absolute system composed of 
part-functions has also the property 
that if a variable changes from inactive 
to active, then amongst the variables 
which affect that variable directly 
there must, at that moment, have 
been at least one which was active. 
One might say, more vividly but 
less accurately, that activity in one 
variable can be obtained only from 
activity in others. A proof is given 
in S. 24/16, but the reason is not 
difficult to see. Suppose for simplicity that a variable A is 
directly affected only by B and C, so that the diagram of 
immediate effects is 

B C 




Figure 14/15/1. In the dif- 
ferent stages the active 
variables are : A, y ; B, y 
and z; C, z; D, x\ E, y; 
F, x and 2. 



Suppose that over a finite interval of time all three have been 
constant, and that the whole is absolute. If B and C remain 
at these constant values, and if A is started at the same value 
as before, then by the absoluteness A's behaviour must be the 
same as before, i.e. A must stay constant. The property has 
nothing to do with energy or its conservation ; nor does it attempt 
to dogmatise about what real 4 machines ' can or cannot do ; it 
simply says that if B and C remain constant and A changes from 
inactive to active, then the system cannot be absolute — in other 
words, it is not completely isolated. 

The sparks which wander in charred paper give a vivid picture 
of this property : they can spread, one can become multiple, or 
several can converge ; but no spark can arise in an unburning 
region. 

14/16. Part-functions were introduced primarily in the hope 
that they would provide a system more readily stabilised than 
one of full-functions. It can now be shown that this is so. 

164 



CONSTANCY AND INDEPENDENCE 14/16 

First, what do we mean by ' difficulty of stabilisation ' ? 
Consider an engineer designing, on the bench, an electronic 
system. He has before him an apparatus which he wants to 
be stable at some particular state. The apparatus contains a 
number of adjustable constants, parameters, and he has to find 
a combination of values that will give him what he wants. The 
4 difficulty ' of stabilisation may be defined as, and measured 
by, the proportion of all possible parameter-values that fail to 
give the required stability. The definition has the advantage 
that it is directly applicable to the homeostat and any similar 
mechanism that has to search through combinations of values. 

With this definition it can be shown that if a system of N 
part-functions has on the average k of its variables active, then 
its difficulty of stabilisation is the same, other things being equal, 
as that of a system of k full-functions. 

The proof is given in S. 24/18, but the theorem is clearly 
plausible. When a system of part-functions is in a region of 
the phase-space where k variables are active and where all the 
other variables are constant, the k variables form a system which 
is absolute and which is not essentially different from any other 
absolute system of k variables. The fact that we have been 
thinking of it differently does not affect the intrinsic nature of 
the situation. Equally, whenever we have postulated an abso- 
lute system, we have assumed that its surrounding variables are 
constant, at least for the duration of the experiment or observa- 
tion. Yet these surrounding variables are usually not constant 
for ever. So our ' absolute system ' was quite commonly only 
a portion of a larger system of part-functions. There is there- 
fore no intrinsic difference between an absolute system of k 
full-functions and a subsystem of k active variables within a 
larger system of part-functions. That being so, there is no reason 
to expect any difference in their difficulties of stabilisation. 

The theorem is of great importance to us, for it means that 
the time taken to stabilise a system of N part-functions will, 
very roughly, be more like T 2 of S. 12/3 than T x ; so the change 
to part-functions may change the stabilisation from ' impossible ' 
to ' possible '. The subject will be developed in S. 17/3. 



165 



CHAPTER 15 

Dispersion 

15/1. Systems of part-functions have the fundamental property 
that each line of behaviour may leave some of the variables 
inactive. Dispersion occurs when the set of variables made 
active by one line of behaviour differs from the set made active 
by another. We will begin to consider the physiological applica- 
tions of this fact. 

First consider a system of full-functions. Suppose we record 
a few of its variables' behaviours while it traverses first one line 
of behaviour and then another. The records would show the 
variables always fluctuating, and the two records would differ 
only in their patterns of fluctuation. Now suppose we have a 
system of part-functions. Again we record some of the variables' 
behaviours. It may happen that from one initial state the line 
of behaviour leaves all the recorded variables inactive, while the 
line from another shows some activity. Since, by S. 6/6, the 
change of initial state corresponds to ' applying a stimulus ', a 
by-standing physiologist would describe the affair as a simple 
case of a mechanism ' responding ' to a stimulus. Since living 
organisms' responses to stimuli have been sometimes offered as 
proof that the organism has some power not possessed by 
mechanisms, we must examine these reactions more closely. 

In S. 6/6 we saw that the most general representation of a 
1 stimulus ' was a change from one initial state to another. Now 
in general, even though the system is absolute, the course of 
the line of behaviour from one initial state puts no restriction 
on the course from another initial state. From this lack of 
restriction follow several consequences. 

15/2. The first consequence is that, in a system known only 
to be complex, however small the difference between the initial 
states — however slight or simple the stimulus — we can put no 
limit to the greatness of the difference between the subsequent 

166 



DISPERSION 15/3 

lines of behaviour. Thus Pavlov conditioned a dog so that it 
gave no salivary response when subjected to the compound 
stimulus of : 

the experimental room, the harness, the feeding apparatus, 
the sound of a metronome beating at 104 per minute, and 
the sound of a No. 16 organ pipe, 
but gave a positive salivary response when subjected to : 

the experimental room, the harness, the feeding apparatus, 
the sound of a metronome beating at 104 per minute, and 
the sound of a No. 15 organ pipe. 
Such a ' discrimination ' has been considered by some to be beyond 
the powers of mechanism, but this is not so : all that is neces- 
sary is that the system should be complex and should contain 
part-functions. 

15/3. The same point of view helps to make clear the physio- 
logical concept of ' adding ' stimuli. In the simple case it is 
easy enough to see what is meant by the ' addition ' of two 
stimuli. If a dog has developed one response to a flashing light 
and another to a ticking metronome, it is easy to apply simul- 
taneously the flashes and the ticks and to regard this stimulation 
as the c sum ' of the two stimuli. But the application of such 
' sums ' was found in many cases to lead to no simple addition 
of responses : a dog could easily be conditioned to salivate to 
flashes and to ticks and yet to give no salivation when both were 
applied simultaneously. Some physiologists have been surprised 
that this could happen. Let us view the events in phase-space. 
Suppose our system has variables a, b, c, . . . and that the 
basal, c control ', behaviour follows the initial state a Q9 b , c , . . . 
Suppose the effect of stimulus A corresponds to the line of 
behaviour from the initial state a l9 b , c , . . . , and that of 
stimulus B to the line from a , b l9 c 09 . . . Then the behaviour 
after the initial state a l9 b lt c , . . . would correspond to the 
response to the simultaneous presentation of A and B. If we 
know the behaviours after A and after B separately, what can 
we predict of the behaviour after their presentation simultane- 
ously ? The possibility is illustrated in Figure 15/3/1, which 
shows at once that the lines of behaviour from II and J in no 
way restrict that from K, which represents, in this scheme, the 
4 sum ' response. 

167 



15/3 DESIGN FOR A BRAIN 

It will be seen, therefore, that in a complex system, every 
group of stimuli will have a holistic quality, in that the response 
to the whole group will not be predictable from the responses 
to the separate stimuli, or even to sub-groups. The dog that 




salivated to each of two stimuli but not to the two together is 
therefore behaving in no way surprisingly, and such behaviour 
is no evidence of any ' supra-mechanistic ' power. In complex 
systems such non-additive compoundings are to be expected. 

15/4. Another variation in stimulus-giving occurs when a 
pattern is varied in some mode of presentation without the 
pattern itself being changed, as when an equilateral triangle is 
shown both erect and inverted. The same argument as before 
prevents us from expecting any necessary relation between the 
two evoked responses. 

In some cases the two evoked responses are found to be the 
same, and to be characteristic of the particular pattern even 
though its presentation may have been much changed : an 
object may be recognised though its image falls on a part of 
the retina never before stimulated by it. This power of Gestalt- 
recognition was also sometimes thought to demand ' supra- 
mechanistic ' powers. But in 1947 Pitts and McCulloch showed 
that any mechanism can show such recognition provided it can 
form an invariant over the group of equivalent patterns. As 
the formation of such invariants demands nothing that cannot 
be supplied by ordinary mechanism, the subject need not be 
discussed further here. 

To sum up, these examples have shown that no matter how 
small the difference between stimuli, or initial states, we can, 
in general, if the system is complex, put no limits to the differ- 
ence that may occur between the subsequent lines of behaviour. 
From this we may deduce that if the system is one with many 

168 



DISPERSION 15/5 

part-functions, we can put no limit to the difference there may 
be between the two sets of variables made active in the two 
responses ; or in other words, there is, in general, no limit to 
the degree of dispersion that may occur other than that imposed 
by the finiteness of the mechanism. 

15/5. It will be proposed later that dispersion is used widely 
in the nervous system. First we should notice that it is used 
widely in the sense-organs. The facts are well known, so I can 
be brief. 

The fact that the sense-organs are not identical enforces an 
initial dispersion. Thus if a beam of radiation of wave-length 
0-5 jli is directed to the face, the eye will be stimulated but not 
the skin ; so the optic nerve will be excited but not the trigeminal. 
But if the wave-length is increased beyond 0-8^, the excitation 
changes from the optic nerve to the trigeminal. Dispersion has 
occurred because a change in the stimulus has moved the excita- 
tion (activity) from one set of anatomical elements (variables) 
to another. 

The sense of taste depends on four histologically-distinguishable 
types of receptors each sensitive to only one of the four qualities 
of salt, sweet, sour, and bitter. If change from one solution to 
another changes the excitation from one type of receptor to 
another, then dispersion has occurred. 

In the skin are histologically-distinguishable receptors sensitive 
to touch, pain, heat, and cold. If a needle on the skin is changed 
from lightly touching it to piercing it, the excitation is shifted 
from the ' touch ' to the 4 pain ' type of receptor ; i.e. dispersion 
occurs. 

In the cochlea, sounds differing in pitch vibrate different parts 
of the basilar membrane. As each part has its own sensitive 
cells and its own nerve-fibres, a change in pitch will shift the 
excitation from one set of fibres to another. 

The three semicircular canals are arranged in planes mutually 
at right-angles, and each has its own sensitive cells and nerve- 
fibres. A change in the plane of rotation of the head will there- 
fore shift the excitation from one set of fibres to another. 

Whether a change in colour of a stimulating light changes 
the excitation from one set of elements in the retina to another 
is at present uncertain. But dispersion clearly occurs when the 

169 



15/6 DESIGN FOR A BRAIN 

light changes its position in space ; for, if the eyeball does not 
move, the excitation is changed from one set of elements to 
another. The lens is, in fact, a device for ensuring that disper- 
sion occurs : from the primitive light-spot of a Protozoon dis- 
persion cannot occur. 

It will be seen therefore that a considerable amount of dis- 
persion is enforced before the effects of stimuli reach the central 
nervous system : the different stimuli not only arrive at the 
central nervous system different in their qualities but they often 
arrive by different paths, and excite different groups of cells. 

15/6. The sense organs evidently have as an important function 
the achievement of dispersion. That it occurs or is maintained 
in the nervous system is supported by two pieces of evidence. 

The fact that cerebral processes, especially those of cellular 
magnitude, frequently show threshold, the fact that this property 
generates part-functions (S. 14/14), and the fact that part- 
functions cause dispersion (S. 14/15) have already been treated. 
The deduction that dispersion must occur within the nervous 
system can hardly be avoided. 

More direct evidence is provided by the fact that, in such cases 
as are known, the tracts from sense-organ to cortex at least 
maintain such dispersion as has occurred in the sense organ. 
The point-to-point representation of the retina on the visual 
cortex, for instance, ensures that the dispersion achieved in the 
retina will at least not be lost. Similarly the point-to-point 
representation now known to be made by the projection of the 
auditory nerve on the temporal cortex ensures that the dispersion 
due to pitch will also not be lost. There are therefore strong 
reasons for believing that dispersion plays an important part in 
the nervous system. What that part is will be discussed in the 
next three chapters. 

Reference 

Pitts, W., and McCulloch, W. S. How we know universals : the pereeption 
of auditory and visual forms. Bulletin of mathematical Biophysics, 
9, 127 ; 1947. 



170 



CHAPTER 16 

The Multistable System 



16/1. The systems discussed in the previous chapter contained 
no step-functions, and the effect of ultrastability on their pro- 
perties was not considered. In this chapter, ultrastability will 
be re-introduced, so we shall now consider what properties will 
be found in systems which show both ultrastability and dispersion. 

To study the interactions of these two properties we might 
start by examining the properties of an ultrastable system whose 
main variables are all part-functions. But it has been found 
simpler to start by considering a system defined thus : a multi- 
stable system consists of many ultrastable systems joined main 
variable to main variable, all the main variables being p art-functions . 

The restriction to part-functions is really slight, for the part- 
function ranges all the way from the full- to the step-function. 
It will further be noticed that, as the ultrastable, or ' sub- ', 
systems are joined main variable to main variable only, each 
step-function will now be restricted in two ways. The critical 
states which determine whether a particular step-function shall 
change value depend only on those main variables that belong 
to the same subsystem. And when a step -function has changed 
value, the immediate effect is confined to that subsystem to 
which it belongs. In the definition of the ultrastable system 
(S. 8/6) no such limitation was imposed. 

This type of system has been defined, not because it is the 
only possible type, but because the exactness of its definition 
makes possible an exact discussion. When we have established 
its properties, we will proceed on the assumption that other 
systems, far too varied for individual study, will, if they approxi- 
mate to the multistable system in construction, approximate 
to it in behaviour. 

171 



16/2 DESIGN FOR A BRAIN 

16/2. The multistable system is itself ultrastable. The pro- 
position may be established by considering the class of ' all 
ultrastable systems '. Such a class will include every system 
not incompatible with the definition of S. 8/6. It will, for 
instance, contain systems whose main variables are all full-func- 
tions, systems some of whose main variables are part-functions, 
and systems whose main variables are all part-functions (S. 11/8). 
Further, the class will include both those whose step-functions 
are wide in their immediate effects and those whose step-functions 
act directly on only a few main variables. The class will there- 
fore include those systems defined as ' multistable '. 

From this fact it follows that all the properties possessed gener- 
ally by the ultrastable system will be possessed by the multi- 
stable. In particular, the multistable system will reject all 
unstable fields of its main variables but will retain the first 
occurring stable field. In other words, the multistable system 
will ' adapt ' just as will any other ultrastable system. 

On the other hand, the faults discussed in Chapter 11 were 
due to the fact that the systems considered before that chapter 
had main variables which were all full-functions. Now that the 
main variables have become all part-functions we shall find, in 
this and the next two chapters, that the faults have been reduced 
or eliminated. 

16/3. In a multistable system, if no step-function changes in 
value, the main variables, being all part-functions, will form a 
system identical with that discussed in S. 14/15. In particular, 
it will show dispersion : two lines of behaviour will make active 
two sets of variables ; the two sets will usually not be identical, 
and may perhaps have no common member. 

16/4. It is now possible to deduce the conditions that must 
hold if a system, multistable or not, is to be able to acquire a 
second adaptation without losing a first. 

We may view the process in two ways, which are really equi- 
valent. First, I will suppose that we have an ultrastable system 
which can be connected to either of two environments (as Units 
3 and 4 of the homeostat, representing the adapting system, 
might be joined to either Unit 1 or Unit 2, representing the 
two environments). Suppose that the system has been joined 

172 



THE MULTISTABLE SYSTEM 16/5 

to environment a, has adapted to it, and has thus reached a 
terminal field. To record this ' first adaptation ', we disturb a 
slightly in various ways and record the system's responses. Give 
the variables activated in these responses the generic label A. 
Next, remove a, join on environment /?, and allow ultrastability 
to establish a ' second adaptation '. Give the generic label S 
to all step-functions that were changed by this process. Finally, 
remove fi, restore a, and again test the system's responses to 
small disturbances applied to a ; compare these responses with 
those first recorded to see whether the first adaptation has been 
retained or lost. For the responses to he unchanged — -for the first 
adaptation to be retained — it is necessary and sufficient that during 
the responses there should be a wall of null-functions between the 
variables A and the step-functions S. The condition is necessary, 
for if an S is not so separated from an A, then at least one A's 
behaviour will be changed. It is also sufficient, for if the wall 
of constancies is present, then by S. 14/8 the A's are independent 
of the S's, and the *S"s changes will not affect the A's responses. 

(The other way of viewing the process is to allow a parameter 
P to affect the ultrastable system, the two environments being 
represented by two values P' and P". The ' disturbance from a ' 
becomes a transient variation in the value of P. The reader 
can verify that this view leads to the same conclusion.) 

The necessary wall of constancies can be obtained in more 
than one way. Thus, if the system really consisted of two 
permanently unconnected parts, one of which was joined to a 
and the other to {3, then the addition of a second adaptation 
would be possible ; so the present discussion includes the case 
of the iterated ultrastable systems. More interesting now is 
the possibility that the constancies have been provided by part- 
functions, for this enables the connections to be temporary and 
conditional. The multistable system is certainly not incapable 
of so acquiring a second adaptation. The facts that set A will 
often be only a fraction of the whole, that part-functions are 
ubiquitous, and that all step-functions are only local in their 
effects makes the separation of A and S readily possible. 

16/5. As a further step towards understanding the multistable 
system, suppose that we are observing two of the subsystems, 
that their main variables are directly linked so that changes of 

173 



16/5 DESIGN FOR A BRAIN 

either immediately affects the other, and that for some reason 
all the other subsystems are inactive. 

The first point to notice is that, as the other subsystems are 
inactive, their presence may be ignored ; for they become like 
the 4 background ' of S. 6/1. Even if some are active, they can 
still be ignored if the two observed subsystems are separated 
from them by a wall of inactive subsystems (S. 14/8). 

The next point to notice is that the two subsystems, regarded 
as a unit, form a whole which is ultrastable. This whole will 
therefore proceed, through the usual series of events, to a terminal 
field. Its behaviour will not be essentially different from that 
recorded in Figure 8/8/5. If, however, we regard the same 
series of events as occurring, not within one ultrastable whole, 
but as interactions between two subsystems, then we shall observe 
behaviours homologous with those observed when interaction 
occurs between ' animal ' and l environment '. In other words, 
within a multistable system, subsystem adapts to subsystem in exactly 
the same way as animal adapts to environment. Trial and error 
will appear to be used ; and, when the process is completed, the 
activities of the two parts will show co-ordination to the common 
end of maintaining the variables of the double system within the 
region of its critical states. 

Exactly the same principle governs the interactions between 
three subsystems. If the three are in continuous interaction, 
they form a single ultrastable system which will have the usual 
properties. 

As illustration we can take the interesting case in which two 
of them, A and C say, while having no immediate connection 
with each other, are joined to an intervening system B, inter- 
mittently but not simultaneously. Suppose B interacts first with 
A : by their ultrastability they will arrive at a terminal field. 
Next let B and C interact. If B's step-functions, together with 
those of C, give a stable field to the main variables of B and C, 
then that set of J5's step-function values will persist indefinitely ; 
for when B rejoins A the original stable field will be re-formed. 
But if Z?'s set with C's does not give stability, then it will be 
changed to another set. It follows that B's step-functions will 
stop changing when, and only when, they have a set of values 
which forms fields stable with both A and C. (The identity in 
principle with the process described in S. 11/4 should be noted.) 

174 



THE MULTISTABLE SYSTEM 



16/5 



The process can be illustrated on the homeostat. Tliree units 
were connected so that the diagram of immediate effects was 
2 ^± 1 ^±: 3 (corresponding to A, B, and C respectively). To 
separate the effects of 2 and 3 on 1, bars were placed across the 
potentiometer dishes (Figure 8/8/2) of 2 and 3 so that they could 
move only in the direction recorded as downwards in Figure 
16/5/1, while 1 could move either upwards or downwards. 
If 1 was above the central line (shown broken), 1 and 2 inter- 
acted, and 3 was independent ; but if 1 was below the central 
line, then 1 and 3 interacted, and 2 was independent. 1 was 




u jVl ^ m * n 



—\j — \r 



v 



Time 



Figure 16/5/1 : Three units of the homeostat interacting. Bars in the 
central positions prevent 2 and 3 from moving in the direction corre- 
sponding here to upwards. Vertical strokes on U record changes of 
uniselector position in unit 1. 

set to act on 2 negatively and on 3 positively, while the effects 
2 — >- 1 and 3 — > 1 were uniselector-controlled. 

When switched on, at J, 1 and 2 formed an unstable system 
and the critical state was transgressed. The next uniselector 
connections (K) made 1 and 2 stable, but 1 and 3 were unstable. 
This led to the next position (L) where 1 and 3 were stable but 
1 and 2 became again unstable. The next position (M) did 
not remedy this ; but the following position (N) happened to 
provide connections which made both systems stable. The values 
of the step-functions are now permanent ; 1 can interact repeatedly 
with both 2 and 3 without loss of stability. 

It has already been noticed that if A, B and C should form 
from time to time a triple combination, then the step-functions 

175 



16/6 DESIGN FOR A BRAIN 

of all three parts will stop changing when, and only when, the 
triple combination has a stable field. But we can go further 
than that. If A, B and C should join intermittently in various 
ways, sometimes joining as pairs, sometimes as a triple, and 
sometimes remaining independent, then their step-functions will 
stop changing when, and only when, they arrive at a set of 
values which gives stability to all the arrangements. 

Clearly the same line of reasoning will apply no matter how 
many subsystems interact or in what groups or patterns they 
join. Always we can predict that their step-functions will stop 
changing when, and only when, the combinations are all stable. 
Ultrastable systems, whether isolated or joined in multistable 
systems, act always selectively towards those step-function values 
which provide stability ; for the fundamental interaction between 
step-function and stability, the principle of ultrastability described 
in S. 8/5, still rules the process. 

16/6. At the beginning of the preceding section it was assumed, 
for simplicity, that the process of dispersion was suspended, for 
we assumed that the two subsystems interacting remained the 
same two during the whole process. What modifications must 
be made when we allow for the fact that in the multistable system 
the number and distribution of subsystems active at each moment 
fluctuates ? 

It is readily seen that the principle of ultrastability holds equally 
whether dispersion is absent or present ; for the proof of Chapter 8 
was independent of special assumptions about the type of vari- 
able. The chief effect of dispersion is to destroy the individuality 
of the subsystems considered in the previous section. There 
two subsystems were pictured as going through the complex 
processes of ultrastability, their main variables being repeatedly 
active while those of the surrounding subsystems remained 
inactive. This permanence of individuality can hardly occur 
when dispersion is restored. Thus, suppose that a multistable 
system's field of all its main variables is stable, and that its repre- 
sentative point is at a resting state R. If the representative point 
is displaced to a point P, or to Q, the lines from these points will 
lead it back to R. As the point travels back from P to R, sub- 
systems will come into action, perhaps singly, perhaps in com- 
bination, becoming active and inactive in kaleidoscopic variety 

176 

/ 



THE MULTISTABLE SYSTEM 16/7 

and apparent confusion. Travel along the other line, from Q to 
R, will also activate various combinations of subsystems ; and 
the set made active in the second line may be very different from 
that made active by the first. 

In such conditions it is no longer profitable to observe par- 
ticular subsystems when a multi stable system adapts. What 
will happen is that instability, and consequent step-function 
change, will cause combination after combination of subsystems 
to become active. So long as instability persists, so long will 
new combinations arise. But when a stable field arises not 
causing step-functions to change, it will, as usual, be retained. 
If now the multistable system's adaptation be tested by dis- 
placements of its representative point, the system will be found 
to respond by various activities of various subsystems, all co- 
ordinated to the common end. But though co-ordinated in this 
way, there will, in general, be no simple relation between the 
actions of subsystem on subsystem : knowing which subsystems 
were activated on one line of behaviour, and how they interacted, 
gives no certainty about which will be activated on some other 
line of behaviour, or how they will interact. 

Later I shall refer again to ' subsystem A adapting to, or 
interacting with, subsystem B ', but this will be only a form of 
words, convenient for description : it is to be understood that 
what is A and what is B may change from moment to moment. 

16/7. In S. 12/4 it was shown that the division of a system 
into parts reduced markedly the time necessary for adaptation. 
The multistable system, being able to adapt by parts (S. 16/5), 
can adapt by this quicker method. But no reason has yet been 
given why this quicker method should be taken if offered. There 
is, however, a well-known principle which ensures this. 

When changes can occur by two processes which differ in their 
speeds of achievement, the faster process, by depriving the 
slower of material, will convert more material than the slow ; 
and if we imagine the material marked in some way according 
to its mode of change, then the major part of the material will 
bear the mark of the faster process. If the difference between 
the speeds is great, then for practical purposes the slow process 
may not be in evidence at all. The important fact here is that 
we can predict a priori that if the change be examined, it will 

177 



16/7 DESIGN FOR A BRAIN 

be found to occur by the fast process ; and we can make this 
prediction without any reference to the particular physical or 
chemical details of the particular change. 

The principle is well known in chemical dynamics. Thus there 
is a reaction whose initial and final states are described by the 
equation 

6FeCl 2 + KCIO3 + 6HC1 = 6FeCl 3 + KC1 + 3H 2 0. 
There are at least two processes leading from the initial to the 
final states : one corresponding to the reaction (of the thirteenth 
order) as written above, and one composed of a series of reactions 
of low order of which the slowest is the reaction (of the third 
order) 

2FeCl 2 + Cl a = 2FeCl 3 . 
The first is slow, for it has to wait for an appropriate collision 
of thirteen molecules, while the second is fast. We can predict 
that the fast will be preferred ; and direct testing has shown that 
the reaction occurs by the second, and not the first, process. 

From this we may draw several deductions. First, the multi- 
stable system will similarly tend to adapt by its fast rather than 
by its slow process. Secondly, since the fast process, by S. 12/4, 
is that of adaptation by a series of small independent parts, any 
multistable system will behave as if it 4 preferred ' to adapt by 
many small independent adaptations rather than by a few com- 
plex adaptations : it ' prefers ' to adapt piecemeal if this is 
possible. Finally, by using the fast process, the time it takes 
in getting adapted will tend to the moderate T 2 (of S. 12/3) 
rather than to the immoderate T v It is therefore at least partly 
free from the fault of excessive slowness described in S. 11/7. 



178 



CHAPTER 17 

Serial Adaptation 



17/1. We have now reached a stage where we must distinguish 
more clearly between the organism and its environment, for the 
concept of the l multistable ' system clearly refers primarily to 
the nervous system. From now on we shall develop the theme 
that the nervous system is approximately multistable, and that 
it is joined to, or interacts with, an environment. But before 
discussing the events in the nervous system we must be clear on 
what we mean by an ' environment '. So far we have left the 
meaning very open : now we want to know what we mean exactly. 
The question occurs in its most urgent form to the designer of a 
4 mechanical brain ', for if he has designed this successfully he 
still has to decide with what it shall interact : having made a 
model of the brain, he must confront it with a model of the 
environment. What model could represent the environment 
adequately in principle ? 

It seems clear that we can, in general, put no limit to what 
may confront the organism. The last century's discoveries have 
warned us that the universe may be inexhaustible in surprises, so 
we should not attempt to define the environment by some formula 
such as ' that which obeys the law of conservation of energy ', 
for the formula may be obsolete before it is in print. In general, 
therefore, the nature of the environment must be left entirely 
open. 

On the other hand, we may obtain a partial definition of some 
practical use by noticing that the living organism on this earth 
adapts not to the whole universe but to some part of it. It is 
often not unlike the homeostat, adapting to a unit or two within 
its immediate cognisance and ignoring the remainder of the world 
around it. Yet, given a particular organism, especially if human, 
we cannot with certainty point to a single variable in the universe 
and say ' this variable will never affect this organism '. This 

179 



17/1 DESIGN FOR A BRAIN 

possibility makes the homeostat unrepresentative ; for a man 
does not, like a prince in a fairy tale, pass instantaneously from one 
world to another, but has rather a series of environments that are 
interrelated, neither wholly separate nor wholly continuous. We 
are, in fact, led again to consider the properties of a system whose 
connections are fluctuating and conditional — the type encoun- 
tered before in S. 11/8, and therefore treatable by the same 
method. I suggest, therefore, that many of the environments 
encountered on this earth by living organisms contain many part- 
functions. Conversely, a system of part-functions adequately 
represents a very wide class of commonly occurring environments. 
As a confirmatory example, here is Jennings' description of an 
hour in the life of Paramecium, with the part-functions indicated 
as they occur. 

(It swims upwards and) ' . . . thus reaches the surface film.' 

The effects of the surface, being constant at zero throughout the 
depths of the pond, will vary as part-functions. A discontinuity 
like a surface will generate part-functions in a variety of ways. 

1 Now there is a strong mechanical jar — someone throws a 
stone into the water perhaps.' 

Intermittent variations of this type will cause variations of part- 
function form in many variables. 

(The Paramecium dives) ' . . . this soon brings it into water 
that is notably lacking in oxygen.' 

The content of oxygen will vary sometimes as part-, sometimes as 
full-, function, depending on what range is considered. Jennings, 
by not mentioning the oxygen content before, was evidently 
assuming its constancy. 

4 ... it approaches a region where the sun has been . . . 
heating the water.' 

Temperature of the water will behave sometimes as part-, sometimes 
as full-, function. 

(It wanders on) ' . . . into the region of a fresh plant stem 
which has lately been crushed. The plant- juice, oozing out, 
alters markedly the chemical constitution of the water.' 

Elsewhere the concentration of these substances is constant at 
zero. 

4 Other Paramecia . . . often strike together ' (collide). 

180 



SERIAL ADAPTATION 17/1 

The pressure on the Paramecium's anterior end varies as a part- 
function. 

4 The animal may strike against stones.' 
Similar part-functions. 

4 Our animal comes against a decayed, softened, leaf.' 

More part-functions. 

4 . . . till it comes to a region containing more carbon dioxide 
than usual.' 

Concentration of carbon dioxide, being generally uniform with 
local increases, will vary as a part-function. 

4 Finally it comes to the source of the carbon dioxide — a large 
mass of bacteria, embedded in zoogloea.' 

Another part-function due to contact. 

It is clear that the ecological world of Paramecium contains 
many part-functions, and so too do the worlds of most living 
organisms. 

A total environment, or universe, that contains many part- 
functions will show dispersion, in that the set of variables active 
at one moment will often be different from the set active at another. 
The pattern of activity will therefore tend, as in S. 14/15, to be 
fluctuating and conditional rather than invariant. As an animal 
interacts with its environment, the observer will see that the 
activity is limited now to this set, now to that. If one set per- 
sists active for a long time and the rest remains inactive and in- 
conspicuous, the observer may, if he pleases, call the first set 4 the ' 
environment. And if later the activity changes to another set he 
may, if he pleases, call it a 4 second ' environment. It is the 
presence of part-functions and dispersion that makes this change of 
view reasonable. 

An organism that tries to adapt to an environment composed 
largely of part-functions will find that the environment is com- 
posed of subsystems which sometimes have individuality and inde- 
pendence but which from time to time show linkage. The alter- 
nation is shown clearly when one learns to drive a car. The 
beginner has to struggle with several subsystems : he has to learn 
to control the steering-wheel and the car's relation to pavement 
and pedestrian ; he has to learn to control the accelerator and 
its relation to engine-speed, learning neither to race the engine nor 

181 N 



17/2 DESIGN FOR A BRAIN 

to stall it ; and he has to learn to change gear, neither burning 
the clutch nor stripping the cogs. On an open, level, empty road 
he can ignore accelerator and gear and can study steering as if 
the other two systems did not exist ; and at the bench he can learn 
to change gear as if steering did not exist. But on an ordinary 
journey the relations vary. For much of the time the three 
systems 

driver + steering wheel -f . . . 
driver -f- accelerator 4- . . . 
driver -f gear lever -f . . . 

could be regarded as independent, each complete in itself. But 
from time to time they interact. Not only may any two use 
common variables in the driver (in arms, legs, brain) but some 
linkage is provided by the machine and the world around. Thus, 
any attempt to change gear must involve the position of the 
accelerator and the speed of the engine ; and turning sharply 
round a corner should be preceded by a slowing down and by a 
change of gear. The whole system thus shows that temporary and 
conditional division into subsystems that is typical of the whole 
that is composed largely of part-functions. 

17/2. Before supposing that the nervous system, in its con- 
struction and function, resembles the multistable, we may ask 
to what extent the supposition is necessary. S. 9/4 showed the 
necessity for ultrastability ; is the hypothesis of multistability 
equally necessary ? 

Our basic facts and assumptions are now as follows : 

(1) the nervous system adapts by the process of ultrastability 

(S. 9/4), 

(2) it can retain one adaptation during the acquisition of 

another (S. 11/3), 

(3) this independence is not achieved by a division of the 

nervous system into permanently separate parts (S. 11/8), 

(4) no special mechanism is to be postulated for special en- 

vironmental conditions (S. 1/9) : if possible, the variables 
are to be statistically homogeneous. 
Given these, what can be deduced ? 

In the system, label the main variables M and the step-functions 
S. Call those variables immediately affected by the first environ- 

182 



SERIAL ADAPTATION 17/3 

ment, M 1 ; those immediately affected by the second, M 2 ; those 
step-functions which changed during the second adaptation, S 2 ; 
and those main variables that S 2 directly affects, M 3 . It is not 
assumed that the M-classes are exclusive. 

After the step-functions S 2 have changed value, the behaviours 
of the variables M x are unchanged, by postulate 2 ; so M x is in- 
dependent of *S 2 (S. 14/3). But S 2 affects M 3 ; so M x must be 
independent of M 3 (S. 14/5). There must therefore be a wall of 
constancies between them (S. 14/8), which must be only temporary, 
by postulate 3. We can deduce therefore that some of the main 
variables must be part-functions. 

Since M 1 is independent of S 2 , it follows that the step-functions 
S 2 can have no immediate effect on the main variables M v In 
other words, some of the step-functions' immediate effects are 
restricted to a few of the main variables. 

If we now use the fourth postulate, that these particular main 
variables and step-functions are typical, it follows that part- 
functions must be common, and step-functions must usually be 
restricted in the variables they immediately affect. We conclude, 
therefore, that if the nervous system is to show the listed proper- 
ties, the main features of the multistable system are necessary. 

17/3. We can now start to examine the thesis that the nervous 
system is approximately multistable. We assume it to be joined 
to an environment that contains many part-functions, and we 
ask to what extent the thesis can explain not only elementary 
adaptation of the type considered earlier but also the more complex 
adaptations of the higher animals, found earlier to be beyond the 
power of a simple system like the homeostat. 

We may conveniently divide the discussion into stages accord- 
ing to the complexity of the environment. First there is the 
environment that, though perhaps extensive, is really simple, for 
it consists of many parts that are independent, so that they can 
be adapted to separately. Such an environment was sketched 
in Figure 12/1/2. It will be considered in this section. Then 
there is the environment that has some connection between its 
parts but where the adaptation can proceed from one part to 
another, perhaps in some order. It will be considered in the 
remainder of this chapter. Then there is the environment that 
is richly interconnected but in which there is still some transient 

183 



17/3 DESIGN FOR A BRAIN 

subdivision into parts, where there are many subsystems, some 
simple, some complex, acting sometimes independently and some- 
times in conjunction, where an adaptation produced for one part 
of the environment may conflict with an adaptation produced for 
another part, and where the adaptations themselves have to be 
woven into more complex patterns if they are to match the com- 
plex demands of the environment. It will be considered in 
Chapter 18. Beyond this, for completeness, are the environments 
of extreme complexity ; but they hardly need discussion, for at 
the limit they go beyond any possibility of being adapted to — 
at least, in the present state of our knowledge. 

The environment of the first type, that composed of indepen- 
dent parts, would, if joined to a multistable system, form an 
ultrastable whole (S. 16/2). Adaptation will, therefore, tend to 
occur. But as the whole is also multistable the process will show 
modifications. Dispersion will occur, so that at each moment 
only some of the whole system's variables will be active. This 
allows the possibility that though the whole may contain a great 
number of variables yet little subsystems may occur containing 
only a few. A subsystem may become stable before all the rest 
are stable. By the usual rule such stable subsystems will tend 
to be self-preserving. There is therefore the possibility that the 
multistable system will adapt piecemeal, its final adaptation 
resembling that of a collection of iterated ultrastable systems, 
like that of Figure 12/1/2. The present system will, however, 
differ in that the constancies that divide subsystem from sub- 
system are not unalterable but conditional. 

Such a multistable system, having arranged itself as a set 
of iterated systems, will show the features previously noticed 
(S. 12/2) : its adaptation will be graduated ; it can conserve its 
old adaptations while developing new ; and, most important, the 
time taken before all its variables become stabilised will 
be reduced from the impossibly long to the reasonably short 
(S. 12/4). 

This is what may happen ; but will it actually occur ? The 
tendency to adaptation may be persistent, but why should the 
process take the favourable course ? First we notice that as 
adaptation in some form or other is inevitable the only question 
is what form it will take. For simplicity, consider an eight- 
variable environment that can be stabilised either in two inde- 

184 



SERIAL ADAPTATION 17/4 

pendent parts of four variables each or in one of eight. During 
the random changes of trial and error, a field stabilising one of the 
sets of four will occur many times more frequently than will a 
field stabilising all eight (S. 20/12). Such a four, once stabilised, 
will retain its field leaving only the other four to find a stable field. 
Consequently, before the process starts we can predict that the 
eight- variable system is much more likely to arrive at stability by 
a sequence of four and four than by a simultaneous eight. The 
fast process is the more probable (S. 16/7). 

We can predict, therefore, that in general if a multistable 
system adapts to an environment composed of P independent 
parts it will tend to develop P independent subsystems, each 
reacting to one part. The nervous system, if multistable, will 
thus tend to adapt to a fragmented environment by a fragmented 
set of reactions, each complete in itself and having no relation to 
the other reactions. It will do this, not because this way is the 
best but because it must. But even though unavoidable, the 
method is by no means unsuitable. It has the great advantage 
of speed — it reduces to a minimum the dangerous period of 
error-making — and there is no point in the nervous system's 
attempting to integrate the reactions when no integration is 
required. 

17/4. The second degree of complexity occurs when the environ- 
ment is neither divided into independent parts nor united 
into a whole, but is divided into parts that can be adapted to 
individually provided that they are taken in a suitable order and 
that the earlier adaptations are used to promote adaptation later. 
Such environments are of common occurrence. A puppy can 
learn how to catch rabbits only after it has learned how to run : 
the environment does not allow the two reactions to be learned in 
the opposite order. A great deal of learning occurs in this way. 
Mathematics, for instance, though too vast and intricate for one 
all-comprehending flash, can be mastered by stages. The stages 
have a natural articulation which must be respected if mastery is 
to be achieved. Thus, the learner can proceed in the order 
' Addition, long multiplication, . . . ' but not in the order ' Long 
multiplication, addition, . . . ' Our present knowledge of mathe- 
matics has in fact been reached only because the subject contains 
such stage-by-stage routes. 

185 



17/5 DESIGN FOR A BRAIN 

As a clear illustration of such a process I quote from Lloyd 
Morgan on the training of a falcon 

' She is trained to the lure — a dead pigeon . . . — at first with 
the leash. Later a light string is attached to the leash, and 
the falcon is unhooded by an assistant, while the falconer, 
standing at a distance of five to ten yards, calls her by shout- 
ing and casting out the lure. Gradually day after day the 
distance is increased, till the hawk will come thirty yards or 
so without hesitation ; then she may be trusted to fly to the 
lure at liberty, and by degrees from any distance, say a 
thousand yards. This accomplished, she should learn to 
stoop to the lure. . . . This should be done at first only 
once, and then progressively until she will stoop backwards 
and forwards at the lure as often as desired. Next she should 
be entered at her quarry . . . ' 

The same process has also been demonstrated more formally. 
Wolfe and Cowles, for instance, taught chimpanzees that tokens 
could be exchanged for fruit : the chimpanzees would then learn 
to open problem boxes to get tokens ; but this way of getting fruit 
(the 4 adaptive ' reaction) was learned only if the procedure for 
the exchange of tokens had been well learned first. In other 
words, the environment was beyond their power of adaptation 
if presented as a complex whole — they could not get the fruit — 
but if taken as two stages in a particular order, could be adapted to. 

4 . . . the growing child fashions day by day, year by year, a 
complex concatenation of acquired knowledge and skills, adding 
one unit to another in endless sequence ', said Culler. I need not 
further emphasise the importance of serial adaptation. 

17/5. To what process in the multistable system does serial 
adaptation correspond ? It is sullicient if we examine the 
relation of a second adaptation to a first, for a series consists only 
of this primary relation repeated. 

We assume then that the multistable system has learned one 
reaction and that it is now faced with an environment that can 
be adapted to only by the system developing some new reaction 
that uses the old. It is convenient, for simplicity, to assume 
here that the first reaction is no longer able to be disrupted by 
subsequent events. The assumption demands little, for in the 
next chapter we shall examine the contrary assumption ; and 
there is, in fact, some evidence to suggest that, in the mammalian 

186 



SERIAL ADAPTATION 17/5 

brain, step-functions that were once labile may become fixed. 
Duncan, for instance, let rats run through a maze, and at various 
times after the run gave them a convulsion by giving an electric 
shock to the brain. He found that if the shock was given within 
about half an hour of the run, all memory of the maze seemed to 
be lost ; but if the shock was given later, the memory was retained. 
In his words : ' It is suggested that newly learned material under- 
goes a period of consolidation or perseveration. Early in this 
period a cerebral electroshock may practically wipe out the effect 
of learning. The material becomes more resistant to such disrup- 
tion ; at the end of an hour no retroactive effect was found.' 
Such a consolidation could easily occur in the animal brain : 
many proteins, for instance, if kept in unusual ionic conditions 
undergo irreversible changes. But with the details we are hardly 
concerned : we simply assume the possibility. 

If, then, the first learned reaction is unbreakable, the whole 
system becomes simple, at least in principle, for as it is an ultra- 
stable system adapting to a system not subject to step-function 
change (i.e. to the complex of environment and first reaction- 
system acting together), the situation is homologous with that 
already treated in Chapter 9 — the adaptation and ' training ' of 
an ultrastable system by an environment. It is therefore not 
playing with words, but expressing a fundamental parallelism to 
say that, in serial learning, the first reaction-system and the 
environment together i train ' the second. They train it by not 
allowing the second to follow lines of behaviour incompatible 
with their own requirements. 

To see the process in more detail, consider the following example. 
A young animal has already learned how to move about the world 
without colliding with objects. (Though this learning is itself 
complex, it will serve for illustration, and has the advantage of 
making the example more vivid.) This learning process was due 
to ultrastability : it has established a set of step-function values 
which give a field such that the system composed of eyes, muscles, 
skin-receptors, some parts of the brain, and hard external objects 
is stable and always acts so as to keep within limits the mechanical 
stresses and pressures caused by objects in contact with the skin- 
receptors (S. 5/4). The diagram of immediate effects will therefore 
resemble Figure 17/5/1. This system will be referred to as part A, 
the c avoiding ' system. 

187 



17/5 



DESIGN FOR A BRAIN 



As the animal must now get its own food, the brain must 
develop a set of step-function values that will give a field in which 
the brain and the food-supply occur as variables, and which is 
stable so that it holds the blood-glucose concentration within 



BRAIN -«- 



EYES 



SKIN 



MUSCLE S 



-^OBJECTS 



Figure 17/5/1 : Diagram of immediate effects of the ' avoiding ' 
system. Each word represents many variables. 

normal limits (S. 5/6). (This system will be referred to as part B, 
the 4 feeding ' system.) This development will also occur by 
ultrastability ; but while this is happening the two systems will 
interact. 

The interaction will occur because, while the animal is making 
trial-and-error attempts to get food, it will repeatedly meet 
objects with which it might collide. The interaction is very 
obvious when a dog chases a rabbit through a wood. Further, 
there is the possibility that the processes of dispersion may allow 
the two reactions to use common variables. When the systems 
interact, the diagram of immediate effects will resemble Figure 
17/5/2. 



BLOOD 
GLUCOSE" 



-^~ BRAIN -<- 



EYES 



FOOD 
SUPPLY 



V 



SKIN 



-«- 



y 

MUSC LES 



B 



^-OBJECTS 



Figure 17/5/2. 



As the * avoiding ' system A is not subject to further step- 
function changes, its field will not alter, and it will at all times 
react in its characteristic way. So the whole system is equivalent 
to an ultrastable system B interacting with an ' environment ' A. 
B will therefore change its step-function values until the whole 
has a field which is stable and which holds within limits the variable 
(blood-glucose concentration) whose extreme deviations cause the 
step-functions to change. We know from S. 8/10 that, whatever 
the peculiarities of A, B's terminal field will be adapted to them. 

188 



SERIAL ADAPTATION 17/5 

It should be noticed that the seven sets of variables (Figure 
17/5/2) are grouped in one way when viewed anatomically and 
in a very different way when viewed functionally. The anato- 
mical point of view sees five sets in the animal's body and two sets 
in the outside world. The functional point of view sees the whole 
as composed of two parts : an ' adapting ' part B, to which A 
is ' environment '. 

It is now possible to predict how the system will behave after 
the above processes have occurred. Because part A, the ' avoid- 
ing ' system, is unchanged, the behaviour of the whole will still 
be such that collisions do not occur ; and the reactions to the 
food supply will maintain the blood-glucose within normal limits. 
But, in addition, because B became adapted to A, the getting of 
food will be modified so that it does not involve collisions, for all 
such variations will have been eliminated. 

If, next, the second reaction becomes unbreakable, by mere 
repetition third and subsequent reactions can similarly be added. 

The multistable system will thus show the phenomenon of serial 
adaptation, not only in its seriality but in the proper adaptation 
of each later acquisition to the earlier. 

References 

Cowles, J. T. Food-tokens as incentives for learning by chimpanzees. 
Comparative Psychology Monographs, 14, No. 71 ; 1937-8. 

Culler, E. A. Recent advances in some concepts of conditioning. Psycho- 
logical Review, 45, 134 ; 1938. 

Duncan, C. P. The retroactive effect of electroshock on learning. Journal 
of comparative and physiological Psychology, 42, 32 ; 1949. 

Wolfe, J. B. Effectiveness of token-rewards for chimpanzees. Compara- 
tive Psychology Monographs, 12, No. 60 ; 1935-6. 



189 



CHAPTER 18 

Interaction between Adaptations 



18/1. At this stage it is convenient to consider in more detail 
the question of ' localisation ' in a multistable system : how will 
the pattern of activity be distributed within it ? In treating the 
brain as a multistable system we followed the incoming sensory 
stimuli through the sense-organs to the sensory cortex (S. 15/5 
and 15/6) ; we have now to consider what happens in the areas of 
4 association ', not of course in detail but sufficiently to develop a 
clear picture of what we would expect to see there. 

Some functions in the cortex are, of course, unquestionably 
localised : the reception of retinal stimuli at the area striata for 
instance. With such I shall not be concerned. I shall consider 
only the localisation of learned reactions, especially of those to 
situations, such as puzzle-boxes containing food, for which the 
organism has no detailed inborn preparation. In such a case the 
simplest hypothesis, the one to be tried first, is that the dispersion 
occurs at random. By this I mean that at each elementary point, 
at each synapse perhaps, the functional details are determined by 
factors of only local significance and action : whether two pieds 
terminaux make contact or three, whether the nucleus happens to 
be on this side of the cell or that, whether five dendrons converge 
or seven (I use these examples only as illustrations of my mean- 
ing). Such details have been determined by primary genetic 
factors modified by merely local incidents in embryological 
development and perhaps by local incidents in past learning. 

But though the local details were once decided by some trifling 
local event, I assume that they persist with some tenacity; for 
learned behaviour can, in the absence of disruptive factors, per- 
sist for many years. I will quote a single example. By differ- 
ential reinforcement with food, Skinner trained twenty young 
pigeons to peck at a translucent key when it was illuminated 
with a complex visual pattern. They were then transferred to the 

190 



INTERACTION BETWEEN ADAPTATIONS 18/1 

usual living quarters where they were used for no further experi- 
ments but served simply as breeders. Small groups were tested 
from time to time for retention of the habit. 

4 The bird was fed in the dimly-lighted experimental apparatus 
in the absence of the key for several days, during which 
emotional responses to the apparatus disappeared. On the day 
of the test the bird was placed in the darkened box. The 
translucent key was present but not lighted. No responses 
were made. When the pattern was projected upon the key, 
all four birds responded quickly and extensively. . . . This 
bird struck the key within two seconds after presentation of 
a visual pattern that it had not seen for four years, and at 
the precise spot upon which differential reinforcement had 
previously been based.' 

I assume, therefore, that the system's behaviour is locally 
regular, in the sense of S. 2/14. 

How will responses be localised in such a system ? Under- 
standing is easier if we first consider the distribution over a town 
of the chimneys that ' smoke ' when the wind blows from a given 
direction. The smoking or not of a particular chimney will be 
locally determinate ; for a wind of a particular force and direc- 
tion, striking the chimney's surroundings from a particular angle, 
will regularly produce the same eddies, which will regularly deter- 
mine the smoking or not of the chimney. But geographically the 
smoking chimneys are distributed more or less at random ; for if 
we mark a plan of the town with a black dot for every chimney that 
smokes in a west wind, and a red dot for every one that smokes 
in a north wind, and then examine the j^lan, we shall find the 
black and red dots intermingled and scattered irregularly. The 
phenomenon of ' smoking '. is thus localised in detail yet dis- 
tributed geographically at random. 

Such is the 4 localisation ' shown by the multistable system. 
We are thus led to expect that the cerebral cortex will show a 
' localisation ' of the following type. The events in the environ- 
ment will provide a continuous stream of information which will 
pour through the sense organs into the nervous system. The set 
of variables activated at one moment will usually differ from 
the set activated at a later moment ; for in this system there 
is nothing to direct all the activities of one reaction into one set 
of variables and all those of another reaction into another set. 
On the contrary, the activity will spread and wander with as 

191 



18/1 DESIGN FOR A BRAIN 

little orderliness as the drops of rain that run, joining and separat- 
ing, down a window-pane. But though the wanderings seem 
disorderly, the whole is regular ; so that if the same reaction is 
started again later, the same initial stimuli will meet the same 
local details, will develop into the same patterns, which will 
interact with the later stimuli as they did before, and the behaviour 
will consequently proceed as it did before. 

This type of system would be affected by removals of material 

in a way not unlike that demonstrated by many workers on the 

cerebral cortex. The works of Pavlov and of Lashley are typical. 

Pavlov established various conditioned reflexes in dogs, removed 

various parts of the cerebral cortex, and observed the effects on 

the conditioned reflexes. Lashley taught rats to run through 

mazes and to jump to marked holes, and observed the effects of 

similar operations on their learned habits. The results were 

complicated, but certain general tendencies showed clearly. 

Operations involving a sensory organ or a part of the nervous 

system first traversed by the incoming impulses are usually 

severely destructive to reactions that use that sensory organ. 

Thus, a conditioned reflex to the sound of a bell is usually abolished 

by destruction of the cochleae, by section of the auditory nerves, 

or by ablation of the temporal lobes. Equally, reactions involving 

some type of motor activity are apt to be severely upset if the 

centre for this type of motor activity is damaged. But it was 

found that the removal of cerebral cortex from other parts of the 

brain gave vague results. Removal of almost any part caused 

some disturbance, no matter from where it was removed or what 

type of reflex or habit was being tested ; and no part could be 

found whose removal would destroy the reflex or habit specifically. 

These results have offered great difficulties to many theories of 

cerebral mechanisms, but are not incompatible with the theory 

put forward here. For in a large multistable system the whole 

reaction will be based on step-functions and activations that are 

both numerous and widely scattered. And, while any exact 

statement would have to be carefully qualified, we can see that, 

just as England's paper-making industry is not to be stopped by 

the devastation of any single county, so a reaction based on 

numerous and widely scattered elements will tend to have more 

immunity to localised injury than one whose elements are few and 

compact. 

192 



INTERACTION BETWEEN ADAPTATIONS 18/3 

18/2. Lashley had noticed this possibility in 1929, remarking 
that the memory-traces might be localised individually without 
conflicting with the main facts, provided there were many traces 
and that they were scattered widely over the cerebral cortex, 
unified physiologically but not anatomically. He did not, how- 
ever, develop the possibility further ; and the reason is not far 
to seek when one considers its implications. 

Such a localisation would, of course, be untidy ; but mere 
untidiness as such matters little. Thus, in a car factory the spare 
parts might be kept so that rear lamps were stored next to radia- 
tors, and ash-trays next to grease guns ; but the lack of obvious 
order would hardly matter if in some way every item could be 
produced when wanted. More serious in the cortex are the effects 
of adding a second reaction ; for merely random dispersion provides 
no means for relating their locations. It not only allows related 
reactions to activate widely separated variables, but it has no 
means of keeping unrelated reactions apart : it even allows them 
to use common variables. We cannot assume that unrelated 
reactions will always differ sufficiently in their sensory forms to 
ensure that the resulting activations stay always apart, for two 
stimuli may be unrelated yet closely similar. Nor is the differen- 
tiation trivial, for it includes the problem of deciding whether a 
few vertical stripes in a jungle belong to some reeds or to a tiger. 

Not only does dispersion lead to the intermingling of sub- 
systems, with abundant chances of random interaction and con- 
fusion, but even more confusion is added with every fresh act 
of learning. Even if some order has been established among the 
previous reactions, each addition of a new reaction is preceded 
by a period of random trial and error which will necessarily cause 
the changing of step-functions which were already adjusted to 
previous reactions, which will be thereby upset. At first sight, 
then, such a system might well seem doomed to fall into chaos. 
Nevertheless, I hope to show that there are good reasons for 
believing that its tendency will actually be towards ever-increasing 
adaptation. 

18/3. Before considering these reasons we should notice that 
the tendency for new learning to upset old is by no means unknown 
in psychology ; and an examination of the facts shows that the 
details are strikingly similar to those that would be expected to 

193 



18/4 DESIGN FOR A BRAIN 

occur if the nervous system were multistable. Pavlov, for in- 
stance, records that ' . . . the addition of new positive, and 
especially of new negative, reflexes exercises, in the great majority 
of cases, an immediate, though temporary, influence upon the 
older reflexes '. And in experimental psychology ' retroactive 
inhibition ' has long been recognised. The evidence is well 
known and too extensive to be discussed here, so I will give 
simply a typical example. Miiller and Pilzecker found that if 
a lesson were learned and then tested after a half-hour interval, 
those who passed the half-hour idle recalled 56 per cent of 
what they had learned, while those who filled the half-hour with 
new learning recalled only 26 per cent. Hilgard and Marquis, 
in fact, after reviewing the evidence, consider that the phenomenon 
is sufficiently ubiquitous to justify its elevation to a ' principle of 
interference '. There can therefore be no doubt that the pheno- 
menon is of common occurrence. New learning does tend to 
destroy old. 

In this the nervous system resembles the multistable ; but the 
resemblance is even closer. In a multistable system, the more the 
stimuli used in new learning resemble those used in previous learn- 
ing, the more will the new tend to upset the old ; for, by the 
method of dispersion assumed here, the more similar are two 
stimuli the greater is the chance that the dispersion will lead them 
to common variables and to common step-functions. In psycho- 
logical experiments it has repeatedly been found that the more the 
new learning resembled the old the more marked was the inter- 
ference. Thus Robinson made subjects learn four-figure numbers, 
perform a second task, and then attempt to recall the numbers ; 
he found that maximal interference occurred when the second task 
consisted of learning more four-figure numbers. Similarly Skaggs 
found that after learning five-men positions on the chessboard, the 
maximal failure of memory was caused by learning other such 
arrangements. The multistable system's tendency to be dis- 
organised by new reactions is thus matched by a similar tendency 
in the nervous system. 

18/4. One factor tending always to lessen the amount of inter- 
action between subsystems is c habituation ', already shown in 
Chapter 13 to be an inevitable accompaniment of an ultrastable 
system's activities. There it was shown that an ultrastable 

194 



INTERACTION BETWEEN ADAPTATIONS 18/5 

system, coupled to a source of disturbance, tends to change its 
step-functions to such values as will render it independent of the 
source. Such a change must also occur if one part of a multis table 
system is repeatedly disturbed by another part ; for the reacting 
system possesses the essential properties, and the origin of the 
disturbance is irrelevant. A subsystem is safe from such dis- 
turbance when and only when its variables are independent of all 
the other variables in the system. There is no necessity for me to 
repeat the evidence here, for it is identical with that of Chapter 13. 
It can therefore be predicted that as the various subsystems of a 
multistable system act on one another, the tendency will be, as 
time goes on, for the various subsystems to upset each other less 
and less. 

If the nervous system is multistable it would show the same 
tendency. It would thus show habituation twice : once in its 
interactions with its environment and again between its various 
component subsystems. Such ' intracerebral ' habituation will 
tend to lessen the disturbing actions of part on part, and it will 
therefore contribute to lessening the chaos described in S. 18/2. 
But such a process will not always lead to complete adaptation ; 
for its tendency, being always to remove interaction, is to divide 
the whole into many independent parts. With some simple 
environments such subdivision may be sufficient, as was noticed 
in S. 17/3 ; but it contributes nothing towards the co-ordination 
of reactions when a complex environment can be controlled only 
by an intricate co-ordination in the nervous system. 

18/5. In the turbulence of many subsystems interacting, the 
principle of ultrastability still holds and still acts persistently in 
the direction of tending to improve the organism's adaptation 
to its environment. It will still act selectively towards the 
useful interactions. Suppose first that two subsystems interact 
in such a way that, though individually adaptive, their com- 
pounded reactions are non-adaptive/ A kitten, for instance, has 
already learned that when it is cold it should go right up to the 
warmth of its mother, and that when it is hungry it should go 
right up to the redness of a piece of meat. If later, when it is 
both cold and hungry, it sees a fire, it would probably tend, in the 
absence of other factors, to go right up to it. But the very fact 
that the interaction leads to non-adaptive behaviour provides the 

195 



18/6 DESIGN FOR A BRAIN 

cause for its own correction : step-functions change value, and that 
particular form of interaction is destroyed. Then the step-func- 
tions' new values provide new forms of interaction, which are again 
tested against the environment. The process can stop when and 
only when the step-functions have values that, acting with the 
environment, give behaviour that keeps the essential variables 
within normal limits. Interactions are thus as subject to the 
requirements of ultrastability as are the other characteristics of 
behaviour. 

Ultrastability thus works in all ways towards adaptation. 
The only question that remains is whether it is sufficiently effective. 

18/6. Is the principle of ultrastability really sufficient to over- 
come the tendency to chaos ? Is it really sufficient to co-ordinate 
the activities of, say, 10 10 neurons when they interact with an 
extremely complicated environment ? Let me admit at once that 
the problem will require a great deal of further study before a final 
answer can be given. The mathematical study of such systems 
has yet hardly begun, so no rigorous proof can be given. The 
available physiological evidence is slight, and the physiologist 
who tries to get direct evidence will encounter formidable diffi- 
culties. Nevertheless, we are not wholly without evidence on the 
subject. 

Consider first the spinal reflexes. If we examine a mammal's 
reflexes, examining them in relation to its daily life, we shall 
usually find, not only that each individual reflex is adapted to the 
environment but that the various reflexes are so co-ordinated in 
their interactions that they work together harmoniously. Nor 
is this surprising, for species whose reflexes are badly co-ordinated 
have an obviously diminished chance of survival. The principle 
of natural selection has thus been sufficient to produce not only 
well-constructed reflexes but co-ordination between them. 

A second example is given by the many complex biochemical 
processes that must be co-ordinated successfully if an organism is 
to live. Not only must a complicated system like the Krebs' 
cycle, involving a dozen or more reactions, be properly co-ordinated 
within itself, but it must be properly co-ordinated into all the other 
cycles and processes with which it may interact. Biochemists 
have already demonstrated something of the complexity of these 
systems and the future will undoubtedly reveal more. Yet in 

196 



INTERACTION BETWEEN ADAPTATIONS 18/6 

the normal organism natural selection has been sufficient to 
co-ordinate them all. 

If now from the principle of natural selection we remove all 
reference to its cytological details, there remains a process strik- 
ingly similar in the abstract to that impelled by the principle of 
ultrastability. Thus natural selection co-ordinates the reflexes 
by repeated application of the two operations : 

(1) test the organism against the environment ; if harmful 

interactions occur, remove that organism ; 

(2) replace it by new organisms, differing randomly from the 

old. 
And it is known that in general these rules are sufficient, given 
time, to achieve the co-ordination. Similarly, the principle of 
ultrastability leads to the repeated application of the two 
operations : 

(1) test the organism against the environment ; if a harmful 

interaction occurs, change the values of the step-functions 
responsible for it ; 

(2) let the new values provide new forms of behaviour, differing 

randomly from the old. 
The analogy between genes and step-functions is most interesting 
and could be developed further ; but it must not distract us now 
The point at issue is : if natural selection's method of action is 
sufficient to account for the co-ordination between spinal reflexes, 
may not ultrastability's method of action also be sufficient to 
account for the co-ordination between cerebral responses, consider- 
ing that the two processes are abstractly almost identical ? 

It may be objected that the spinal reflexes do not have to fear 
disorganisation by new learning ; but the objection will not stand. 
We are comparing the ontogenetic progress of the cerebral re- 
sponses with the phylogenetic progress of the spinal reflexes. As 
the species evolves, its environment changes and new reflexes have 
to be developed to suit the new conditions. Each new reflex, 
though suitable in itself, may cause difficulties if it compounds 
badly with the pre-existing reflexes. Thus, a bird that developed 
a new reflex for pecking at a new type of white, round, edible 
fungus might be in danger of using the same reflex on its eggs. 
Evolution has thus often had to face the difficulty that a harmoni- 
ous set of reflexes will be disorganised if an extra reflex is added. 
Again, consider the biochemical systems. As most, if not all, 

197 o 



18/6 DESIGN FOR A BRAIN 

genes have biochemical effects, the acquisition of many a new 
favourable mutation has meant that a harmonious set of bio- 
chemical reactions has had to be reorganised to allow the incor- 
poration of the new reaction. 

Evolution has thus had to cope, phylogenetically, with all the 
difficulties of integration that beset the individual ontogenetically. 
The tendency to ' chaos ', described in S. 18/2, thus occurs in the 
species as well as in the individual. In the species, the co-ordinat- 
ing power of natural selection has shown itself stronger than the 
tendency to chaos. 

Natural selection is effective in proportion to the number of 
times that the selection occurs : in a single generation it is negli- 
gible, over the ages irresistible. And if the unrepeated action of 
ultrastability seems feeble, might it not become equally irresistible 
if the nervous system was subjected to its action on an equally 
great number of occasions ? 

How often does it act in the life of, say, the average human 
being ? I suggest that in those reactions where interaction is 
important and extensive, the total duration of the learning process 
is often of ' geological ' duration when compared with the duration 
of a single incident, in that the total number of incidents contri- 
buting to the final co-ordination is very large. I will give a single 
example. Consider the adult's ability to make a prescribed move- 
ment without hitting a given object — to put the cap on a fountain- 
pen, say, without damaging the nib. This skill demands co- 
ordinated activity, but the co-ordination has not been developed 
by a single experience. Here are some of the incidents that will 
probably have contributed to this particular skill : 

Putting the finger into the mouth (without hitting lips or teeth) ; 
putting the finger into the handle of a cup (without striking the 
handle) ; dipping pen into inkwell (without striking the rim) ; 
putting button into button-hole ; passing a shoe-lace through its 
hole ; inserting a collar-stud into the neck-band ; putting pen-nib 
into pen ; making a knot by passing an end through a loop ; 
replacing the cork in a bottle ; putting a key into a key-hole ; 
threading a needle ; placing a gramophone record on the turn- 
table's central pin ; putting the finger into a ring ; inserting a 
funnel into a flask ; putting a cigarette into a holder ; putting 
a cuff-link into a cuff ; inserting a pipe-cleaner ; putting a screw 
into a nut ; and so on. 

198 



INTERACTION BETWEEN ADAPTATIONS 18/6 

Not only could the list be extended almost indefinitely, but 
each item is itself representative of a great number of incidents, 
carried out on a variety of occasions in a variety of ways. The 
total number of incidents contributing to the adult's skill may thus 
be very large. 

So by the time a human being has developed an adult's skill 
and knowledge, he has been subjected to the action of ultrastability 
repetitively to a degree which may be comparable with that to 
which an established species has been subjected to natural selec- 
tion. If this is so, it is not impossible that ultrastability can 
account fully for the development of adaptive behaviour, even 
when the adaptation is as complex as that of Man. 

References 

Ashby, W. Ross. Statistical machinery. Thales, 7, 1 ; 1951. 

Lashley, K. S. Nervous mechanisms in learning. The foundations of 
experimental psychology, edited C. Murchison. Worcester, 1929. 

Muller, G. E., and Pilzecker, A. Experimentelle Beitrage zur Lehre 
vom Gedachtniss. Zeitschrift fur Psychologie und Physiologie der Sinnes- 
organe, Erganzungsband No. 1 ; 1900. 

Robinson, E. S. Some factors determining the degree of retroactive inhibi- 
tion. Psychological Monographs, 28, No. 128 ; 1920. 

Skaggs, E. B. Further studies in retroactive inhibition. Psychological 
Monographs, 34, No. 161 ; 1925. 

Skinner, B. F. Are theories of learning necessary ? Psychological Review, 
57, 193 : 1950. 



199 



APPENDIX 



CHAPTER 19 

The Absolute System 



(Some of the definitions already given are re- 
peated here for convenience) 

19/1. A system of n variables will usually be represented by 
x v . . . , x n , or sometimes more briefly by x. n will be assumed 
finite ; a system with an infinite number of variables (e.g. that 
of S. 19/23), where xi is a continuous function of i, will be 
replaced by a system in which i is discontinuous and n finite, 
and which differs from the original system by some negligible 
amount. 

19/2. Each variable x% is a function of the time t ; it will 
sometimes be written as xi(t) for emphasis. It must be single- 
valued, but need not be continuous. A constant may be 
regarded as a variable which undergoes zero change. 

19/3. The state of a system at a time t is the set of numerical 
values of x^t), . . . , x n (t). Two states are ' equal ' if n equalities 
exist between the corresponding pairs. 

19/4. A line of behaviour is specified by a succession of states 
and the time-intervals between them. Two lines of behaviour 
which differ only in the absolute times of their initial states are 
equal. 

19/5. A geometrical co-ordinate space with n axes x v . . . , x n , 
and a dynamic system with variables x v . . . , x n provide 
a one-one correspondence between each point of the space 
(within some region) and each state of the system. The region 
is the system's ' phase-space '. 

19/6. A primary operation discovers the system's behaviour by 

203 



19/7 DESIGN FOR A BRAIN 

finding how it behaves after being released from an initial state 
#J, . . . , x„. It generates one line of behaviour. 

The field of a system is its phase-space filled with such lines 
of behaviour. 



19/7. If, on repeatedly applying primary operations to a 
system, it is found that all the lines of behaviour which follow 
an initial state S are equal, and if a similar equality occurs after 
every other initial state S', S", . . . then the system is regular. 
Such a system can be represented by equations of form 

x x = F x (x\ 9 ...,«£; 01 



x n — t n \x^ . . . , x n ; t)J 

Obviously, if the initial state is at t = 0, we must have 

Fi(x° v . . . , x° n ; 0) = a£ N,l n). 

The equations are the written form of the lines of behaviour ; 
and the forms F{ define the field. They are obtained directly 
from the results of the primary operations. 

19/8. If, on repeatedly applying primary operations to a system, 
it is found that all lines of behaviour which follow a state S 
are equal, no matter how the system arrived at S, and if a 
similar equality occurs after every other state S\ S", . . . then 
the system is absolute. 

19/9. A system is ' state-determined ' if the occurrence of a 
particular state is sufficient to determine the line of behaviour 
which follows. Reference to the preceding section shows that 
absolute systems are state-determined, and vice versa. 



The equations of an absolute system form a group 
19/10. Theorem. That the equations 

Xi = Fi{x\ t . . . , a£ ; t) (i = l n) 

should be those of an absolute system, it is necessary that, re- 
garded as a substitution converting #J, . . ., #° to x v . . ., x n , 

204 



THE ABSOLUTE SYSTEM 



19/10 



they should form a finite continuous (Lie) group of order one 
with t as parameter. 

(1) The system is assumed absolute. Let the initial state of 
the variables be x°, where the single symbol represents all n, and 
let time t' elapse so that x° changes to x' . With x' as initial 



x»— 




Figure 19/10/1. 

state let time t" elapse so that x' changes to x". As the system 
is absolute, the same line of behaviour will be followed if the 
system starts at x° and goes on for time f -f- t". So 

x- = Fi{xi . . . , a£ ; t") = Fi(x° v . . . ,' x\ ; t' + t") 

(i = 1, . . 



n) 



but 



Xi 



Fi{x° l9 



,o . 



n 



(* = i, 



., n) 



giving 

FilF^x"; n • .» Fn(af>; t'); t"} 

= Fi{x° v . . . , x Q n ; t' + n (i = 1, . . . , n) 

for all values of x°, f and t" over some given region. The equation 
is known to be one way of defining a one-parameter finite con- 
tinuous group. 

(2) The group property is not, however, sufficient to ensure 
absoluteness. Thus consider x = (1 -f- t)x° ; the times do not 
combine by addition, which has just been shown to be necessary. 

Example : The system with lines of behaviour given by 



Xl = x\ + x\t -f Z 2 1 

Xo — Xo -f- "I J 



is absolute, but the system with lines given by 

x x = x\ + x\t + V 



is not. 



'2 — 2 ~" • 



205 



19/11 DESIGN FOR A BRAIN 

The canonical equations of an absolute system 

19/11. Theorem : That a system x v . . . , x n should be absolute 
it is necessary and sufficient that the #'s, as functions of t, should 
satisfy differential equations 
dx 1 



i — fl\ x l> - • • t x n) 



~rr — JnK^n • • • » x n) 



(i) 



where the /'s are single-valued, but not necessarily continuous, 
functions of their arguments ; in other words, the fluxions of 
the set x v . . . , x n can be specified as functions of that set and 
of no other functions of the time, explicit or implicit. 

(The equations will be written sometimes as shown, sometimes 
as dxi/dt =fi(x v . . . , #») [i == 1, . . . , n) . (2) 

and sometimes abbreviated to x = f(x), where each letter repre- 
sents the whole set, when the context indicates the meaning 
sufficiently.) 

(1) Start the absolute system at x\, . . . , a% at time t = 
and let it change to x v . . . , x n at time t, and then on to 
x x + das l9 . . . , x n + dx n at time t + dt. Also start it at 
SB l9 . . . , x n at time t = and let time dt elapse. By the group 
property (S. 19/10) the final states must be the same. Using 
the same notation as S. 19/10, and starting from a£, Xi changes 
to Fi(x° ; t + dt) and starting at x% it gets to Fi(x ; dt). 
Therefore 

Fi(x° ; t + dt) = Fi{x ; dt) (i = 1, . . . , n). 

Expand by Taylor's theorem and write ^rFi(a ; b) as F'i{a ; b). 

Then 

Fi{x° ; t) + dt.Fl(x° ; t) = Fi{x ; 0) + dt.F'lx ; 0) 

{% = 1, . . . , n) 

But both Fi(x° ; /) and Fi(x ; 0) equal xi. 

Therefore F^x ; t) = F^x ; 0) {i = 1, . . . , n) . (3) 

But Xi = Fi(x° ; t) \i = 1 n) 

206 



THE ABSOLUTE SYSTEM 19/11 

so, by (3), °f t = F,(x ; 0) (* = 1, . . . , n) 

which proves the theorem, since F\(x ; 0) contains t only in 
x v . . . , x n and not in any other form, either explicit or 
implicit. 

Example 1 .* The absolute system of S. 19/10, treated in this 
way, yields the differential equations 

dx x 

~dt 

dx 2 

~dt 

The second system may not be treated in this way as it is not 
absolute and the group property does not hold. 
Corollary : 



— #2 



Ji\Xi, . . . , X n ) = 



■Fi(x v . . . , x n ; t)\ (i = 1, . . . , n) 



dt 

(2) Given the differential equations, they may be written 
dxi =fi{x v . . . , x n ).dt (i = 1, . . . , n) 

and this shows that a given set of values of x v . . . , x n , i.e. 
a given state of the system, specifies completely what change 
dxi will occur in each variable xi during the next time-interval 
dt. By integration this defines the line of behaviour from that 
state. The system is therefore absolute. 
Example 2 : By integrating 

dx 1 

~dt 

dx 2 

dt 
the group equations of the example of S. 19/10 are regained. 

Example 3 : The equations of the homeostat may be obtained 
thus : — If xi is the angle of deviation of the ith. magnet from 
its central position, the forces acting on xi are the momentum, 
proportional to xi, the friction, also proportional to xi, and the 
four currents in the coil, proportional to x v x 2 , x 3 and # 4 . If 
linearity is assumed, and if all four units are constructionally 
identical, we have 

j t (mxi) = — kxi + l(p — q)(a il x 1 + . . . + a^x^ 

(i = 1, 2, 3, 4) 
207 



19/11 DESIGN FOR A BRAIN 

where p and q are the potentials at the ends of the trough, I 
depends on the valve, k depends on the friction at the vane, 
and m depends on the moment of inertia of the magnet. If 

h = * t j = _ then the equations can be written 



m m 

dxi _ . 
~dt~ Xi 

dx~ 
— = h(a il x 1 + . . . + a i4 oj 4 ) — jxi 



(i = 1, 2, 3, 4) 



which shows the 8 -variable system to be absolute. 
They may also be written 

dxi _ . 

dt ~ m\ k {ChlXl + * * * + a ^ l *> 

Let m — > 0. dxi/dt becomes very large, but not dxi/dt. 
So xi tends rapidly towards 

k ^ Xl + • • • + ^4^4) 

while the CD's, changing slowly, cannot alter rapidly the value 
towards which xi is tending. In the limit, 

^ = £i = fcZ-%^ + . . . + aii x t ) (i = 1, 2, 3, 4) 

Change the time-scale by r = , 1 ; 

— = a il x 1 + . . . + ai^ (i = 1, 2, 3, 4) 

showing the system x v . . . , x± to be absolute and linear. The 
a's are now the values set by the hand-controls of Figure 8/8/3. 



19/12. That a system should be absolute, it is necessary and 
sufficient that at no point of the field should a line of behaviour 

208 



THE ABSOLUTE SYSTEM 19/15 

bifurcate. The statement can be verified from the definition or 
from the theorem of S. 19/11. The statement does not prevent 
lines of behaviour from running together. 

19/13. The theorems of the previous four sections show that 
the following properties, collected for convenience, in a system 
x lt . . . , x n , are all equivalent in that the possession of any one 
of them implies the others : 

(1) From any point in the field departs only one line of 

behaviour (S. 19/8) ; 

(2) the system is state-determined (S. 19/9) ; 

(3) the system has lines of behaviour whose equations specify 

a finite continuous group of order one ; 

(4) the system has lines of behaviour specified by differential 

equations of form 

-^ =fi(xv • • • > ®n) (i = 1, • • • , n) 

where the right-hand side contains no functions of t 
except those whose fluxions are given on the left. 

19/14. From the experimental point of view the simplest test 
for absoluteness is to see whether the lines of behaviour are 
state-determined. An example has been given in S. 2/15. It 
will be noticed that experimentally one cannot prove a system 
to be absolute — one can only say that the evidence does not 
disprove the possibility. On the other hand, one value may be 
sufficient to prove that the system is not absolute. 

19/15. A simple example of a system which is regular but not 
absolute is given by the following apparatus. A table top is 
altered so that instead of being flat, it undulates irregularly but 
gently like a putting-green (Figure 19/15/1). Looking down on 
it from above, we can mark across it a rectangular grid of lines 
to act as co-ordinates. If we place a ball at any point and then 
release it, the ball will roll, and by marking its position at, say, 
every one-tenth second we can determine the lines of behaviour 
of the two-variable system provided by the two co-ordinates. 

If the table is well made, the lines of behaviour will be accur- 
ately reproducible and the system will be regular. Yet the 
experimenter, if he knew nothing of forces, gravity, or momenta, 

209 



19/16 DESIGN FOR A BRAIN 

would find the system unsatisfactory. He would establish that 
the ball, started at A, always went to A' ; and started at B it 
always went to B'. He would find its behaviour at C difficult 
to explain. And if he tried to clarify the situation by starting 
the ball at C itself, he would find it went toD! He would say 
that he could make nothing of the system ; for although each 



Figure 19/15/1. 

line of behaviour is accurately reproducible, the different lines 
of behaviour have no relation to one another. 

This lack of relation means that they do not form a ' group '. 
But whether the experimenter agrees with this or not, he will, 
in practice, reject this 2- variable system and will not rest till 
he has discovered, either for himself or by following Newton, 
a system that is state-determined. In my theory I insist on the 
systems being absolute because I agree with the experimenter 
who, in his practical work, is similarly insistent. 

19/16. That the field of a system should not vary with time, 
it is necessary and sufficient that the system be regular. The 
proof is obvious. 

19/17. One reason why a system's absoluteness is important is 
because the system is thereby shown to be adequately isolated 
from other unknown and irregularly varying parameters. This 
demonstration is obviously fundamental in the experimental 
study of a dynamic system, for the proof of isolation comes, 

210 



THE ABSOLUTE SYSTEM 19/20 

not from an examination of the material substance of the system 
(S. 14/1), which may be misleading and in any case presupposes 
that we know beforehand what makes for isolation and what 
does not, but from a direct test on the behaviour itself. 

Closely related to this in a fundamental way is the fact that 
Shannon's concept of a ' noiseless transducer ' is identical in defini- 
tion with my definition of an absolute system. Thus he defines 
such a transducer as one that, having states a and an input 
x y will, if in state a n and given input x n> change to a new state 
a„+i that is a function only of x n and a n : 

a «+l = g(Vn, a n ) 
Though expressed in a superficially different form, this equation 
is identical with my ' canonical ' equation, for it says simply 
that if the parameters x and the state of the system are given, 
then the system's next step is determined. Thus the com- 
munication engineer, if he were to observe the physicist and the 
psychologist for the first time, would say that they seem to 
prefer to work with noiseless systems. His remark would not 
be as trite as it seems, for from it flow far-reaching consequences 
and the possibilities of rigorous deduction. 

19/18. A second feature which makes absoluteness important 
is that its presence establishes, by appeal only to the behaviour, 
that the system of variables is complete, i.e. that it includes all 
the variables necessary for the specification of the system. 

19/19. When we assemble a machine, we usually know the 
canonical equations directly. If, for instance, certain masses, 
springs, magnets, be put together in a certain way the mathe- 
matical physicist knows how to write down the differential 
equations specifying the subsequent behaviour. 

His equations are not always in our canonical form, but they 
can always be converted to this form provided that the system 
is isolated, i.e. not subjected to arbitrary interference, and is 
determinate. 

19/20. In general there are two methods for studying a dynamic 
system. One method is to know the properties of the parts 
and the pattern of assembly. With this knowledge the canonical 

211 



19/21 DESIGN FOR A BRAIN 

equations can be written down, and their integration predicts the 
behaviour of the whole system. The other method is to study 
the behaviour of the whole system empirically. From this 
knowledge the group equations are obtained : differentiation of 
the functions then gives the canonical equations and thus the 
relations between the parts. 

Sometimes systems that are known to be isolated and complete 
are treated by some method not identical with that used here. 
In those cases some manipulation may be necessary to convert 
the other form into ours. Some of the possible manipulations 
will be shown in the next few sections. 

19/21. Systems can sometimes be described better after a change 
of co-ordinates. This means changing from the original variables 
x v . . . , x n to a new set y v . . . , y m equal in number to the 
old and related in some way 

y% = </>i(x l9 ...,#„) (i = 1, . . . , n) 

If we think of the variables as being represented by dials, the 
change means changing to a new set of dials each of which 
indicates some function of the old. It is easily shown that such 
a change of co-ordinates does not change the absoluteness. 

19/22. In the ' homeostat ' example of S. 19/11 a derivative 
was treated as an independent variable. I have found this 
treatment to be generally advantageous : it leads to no difficulty 
or inconsistency, and gives a beautiful uniformity of method. 

For example, if we have the equations of an absolute system 
we can write them as 

& —M®v . . . , as*) = (» = i, . . . , n) 

treating them as n equations in 2n algebraically independent 
variables x v . . . , x n , x v . . . , x n . Now differentiate all the 
equations q times, getting (q + l)n equations with (q + 2)n 
variables and derivatives. We can then select n of these vari- 
ables arbitrarily, and noticing that we also want the next higher 
derivatives of these ?i, we can eliminate the other qn variables, 
using up qn equations. If the variables selected were z l9 . . . , 0» 
we now have n equations, in 2n variables, of type 

&i(z v . . . , z„, z v . . . , i n ) = (f = 1, . . . , n) 

212 



THE ABSOLUTE SYSTEM 19/24 

These have only to be solved for z v . . . , z n in terms of 
z l9 . . . , z n and the equations are in canonical form. So the 
new system is also absolute. 

This transformation implies that in an absolute system we can 
avoid direct reference to some of the variables provided we use 
derivatives of the remaining variables to replace them. 

Example : x l = x x — x 

Xn i)X 1 ~\~ Xaj 

can be changed to omit direct reference to x 2 by using x x as a 
new independent variable. It is easily converted to 



dx x 

~dt ~~ 
dx. 



'1 



which is in canonical form in the variables x x and x v 



19/23. Systems which are isolated but in which effects are 
transmitted from one variable to another with some finite delay 
may be rendered absolute by adding derivatives as variables. 
Thus, if the effect of x x takes 2 units of time to reach a? 2 , while 
x 2 's effect takes 1 unit of time to reach x v and if we write x(t) 
to show the functional dependence, 

then ^=/i(«i(ft «#-»» 



dx 2 (t) 
dt 



= f 2 {x 1 (t - 1), x 2 (t)}. 



This is not in canonical form ; but by expanding x x (t — 1) and 
x 2 (t — 2) in Taylor's series and then adding to the system as 
many derivatives as are necessary to give the accuracy required, 
we can obtain an absolute system which resembles it as closely 
as we please. 



19/24. If a variable depends on some accumulative effect so 

that, say, x t =f\\ <j>{cc 2 )dt>, then if we put <f>{x 2 )dt = y, we get 

213 P 



19/25 DESIGN FOR A BRAIN 

the equivalent form 

^?- etc 

dt — ' ' etC * 

which is in canonical form. 

19/25. If a variable depends on velocity effects so that, for 
instance 

dx 1 _ f dx 2 \ 

dt - Jl \df * v x v 

-jj£ = J2\ X V X 2) 

dx 
then if we substitute for -=-* in/i(. . .) we get the canonical form 

—jr = fxifiPufitl* x v X 2) 

2 S" I \ 

~fa —J2\ X V X 2) 

19/26. If one variable changes either instantaneously or fast 
enough to be so considered without serious error, then its value 
can be given as a function of those of the other variables ; and 
it can therefore be eliminated from the system. 

19/27. Explicit solutions of the canonical equations 

dxi/dt =fi{x v . . . , x n ) (i — 1,- . . . , n) 

will seldom be needed in our discussion, but some methods will 
be given as they will be required for the examples. 

(1) A simple symbolic solution, giving the first few terms of 
x\ as a power series in t, is given by 

an = <* x x\ (t = 1 n) . . (1) 

where X is the operator 

/i«. ■ • • . Ogjg + • • • +/ " ( '* • • • > xl) h n ■ (2) 

and e' x = l+tX+£x*+~X» + . . . . (3) 

214 



THE ABSOLUTE SYSTEM 19/28 

It has the important property that any function @(x lf . . . , x n ) 

can be shown as a function of t, if the aj's start from x®, . . . , x„, 

by 0(x v . . . , x n ) = <^<Z>(^, . . . t 3,0) _ (4) 
(2) If the functions jfi are linear so that 



-5— = ttj^j -f- di2 X 2 l • • • 1" a \n&n ~T "1 



dx n 

dt ' 



— Cl n iX^ -j- ^712^2 1 • • ■ "T" "nn^n 1 ^n 



(5) 



then if the fr's are zero (as can be arranged by a change of 
origin) the equations may be written in matrix form as 

x = Ax . . . (6) 

where x and x are column vectors and A is the square matrix 
[ciij]. In matrix notation the solution may be written 

x = e tA x° .... (7) 
(3) Most convenient for actual solution of the linear form is 
the recently developed method of the Laplace transform. The 
standard text-books should be consulted for details. 

19/28. Any comparison of an absolute system with the other 
types of system treated in mechanics and in thermodynamics 
must be made with caution. Thus, it should be noticed that the 
concept of the absolute system makes no reference to energy or 
its conservation, treating it as irrelevant. It will also be noticed 
that the absolute system, whatever the ' machine ' providing it, 
is essentially irreversible. This can be established either by 
examining the group equations of S. 19/10, the canonical equa- 
tions of S. 19/11, or, in a particular case, by examining the field 
of the common pendulum in Figure 2/15/1. 

Reference 

Shannon, C. E. A mathematical theory of communication. Bell System 
technical Journal, 27, 379-423, 623-56 ; 1948. 



215 



CHAPTER 20 

Stability 



20/1. ' Stability ' is defined primarily as a relation between a 
line of behaviour and a region in phase-space because only in 
this way can we get a test that is unambiguous in all possible 
cases. Given an absolute system and a region within its field, 
a line of behaviour from a point within the region is stable if it 
never leaves the region. 

20/2. If all the lines within a given region are stable from all 
points within the region, and if all the lines meet at one point, 
the system has ' normal ' stability. 

20/3. A resting state can be defined in several ways. In the 
field it is a terminating point of a line of behaviour. In the 
group equations of S. 19/10 the resting state X v . . . , X n is 
given by the equations 

Xi = Lim Fi(x° ; t) [i = 1, . . . , n) . (1) 

t >-00 

if the n limits exist. In the canonical equations the values satisfy 

fi(X l3 . . ., X n )=0 (t = l n) . (2) 

A resting state is an invariant of the group, for a change of t 
does not alter its value. 

m 

dxj 

be symbolised by J, is not identically zero, then there will be 
isolated resting states. If J = 0, but not all its first minors are 
zero, then the equations define a curve, every point of which 
is a resting state. If J = and all first minors but not all second 
minors are zero, then a two-way surface exists composed of 
resting states ; and so on. 



If the Jacobian of the /'s, i.e. the determinant 



which will 



20/4. Theorem : If the /'s are continuous and differentiable, 
an absolute system tends to the linear form (S. 19/27) in the 
neighbourhood of a resting state. 

216 



STABILITY 



20/5 



Let the system, specified by 

dxi/dt =fi( x v • • • » x n) (t = 1, . . . , n) 
have a resting state Xj, . • . , X n , so that 

fi(X v . . . , X n ) = (t = 1 n) 

Put Xi = Xi -f- & (i = 1, . . . , m) so that xi is measured as a 

deviation ft from its resting value. Then 

d 



dt 



(Xi + ft) =f i (X 1 + & x n + ft.) 



(i = 1, . . . , n) 



Expanding the right-hand side by Taylor's theorem, noting that 
dXi/dt = and that/i(Z) = 0, we find, if the £'s are infinitesimal, 
that 



d£i _ dfi dft 

si T" • • • ~r 37?n 



(* = L 



., n) 



* aii * ' ' " ' ' din- 

The partial derivatives, taken at the point X v . . . , X n , are 
numerical constants. So the system is linear. 

20/5. In general the only test for stability is to observe or 
compute the given line of behaviour and to see what happens 
as t — ■> oo. For the linear system, however, there are tests that 
do not involve the line of behaviour explicitly. Since, by the 
previous section, many systems approximate to the linear within 
the region in which we are interested, the methods to be de- 
scribed are widely applicable. 

Let the linear system be 
dxi 
dt 



&%-iX-, 



0>i2p2 T" • • • i Min^n 



(i = i, 



n) (1) 



or, in the concise matrix notation (S. 19/27) 

x = Ax . . . (2) 

Constant terms on the right-hand side make no difference to 
the stability and can be ignored. If the determinant of A is not 
zero, there is a single resting state. The determinant 



'ii 



■X a 



12 



21 



22 ■ 



-;. . 



a n 



when expanded gives a polynomial 



'\n 
' 2 n 



(Inn A. 

in A of degree n which, when 



equated to 0, gives the characteristic equation of the matrix A 



217 



0. 



20/6 



DESIGN FOR A BRAIN 



20/6. Each coefficient rm is the sum of all i-vowed principal 
(co-axial) minors of A, multiplied by (— 1)*. Thus, 

m l = — («11 + «22 + • • • + a nn) \ Wl„ = (— l) n | A |. 

Example : The linear system 

dxjdt = — 5x ± + 4a? 2 — 6^3! 



7a; 1 



6x 2 + 8x 3 
4# 3 , 



dxjdt = 

dx 3 /dt = — 2x x + 4^2 

has the characteristic equation 

A 3 + 15A 2 + 21 + 8 = 



20/7. Of this equation the roots X lt . . . , A B are the latent 

roots of ^4. The integral of the canonical equations gives each 
X{ as a linear function of the exponentials eV, . . . , eV. For 
the sum to be convergent, no real part of A ls . . . , A n must be 
positive, and this criterion provides a test for the stability of 
the system. 

Example : The equation A 3 + 15A 2 -f 2 A + 8 = has roots 
— 14-902 and — 0-049 ± 0-729 V^^T, so the system of the 
previous section is stable. 



20/8. A test which avoids finding the latent roots is Hurwitz' : 
a necessary and sufficient condition that the linear system is 
stable is that the series of determinants 

etc. 



m v 


m x 1 


i 


m 1 1 


, 


m x 1 










m 3 m 2 




in 3 vi 2 m 1 




m z m 2 


m 1 


1 




m 5 7?i 4 m z 




m 5 m± 


m 3 


m 




m 7 m 6 


m 5 


m t 


(where, i 


f q > n, 


m q 


= 0), are all 


po 


sitive. 







Example : The system with characteristic equation 
P + 15A 2 + 2A + 8 = 
yields the series 



+ 15, 



15 

8 



15 

8 





15 

8 



These have the values + 15, + 22, and + 176. 
is stable, agreeing with the previous test. 

218 



So the system 



STABILITY 



20/11 



20/9. If the coefficients in the characteristic equation are not 
all positive the system is unstable. But the converse is not 
true. Thus the linear system whose matrix is 

i V 6 ° 

— V 6 i ° 

—3^ 

has the characteristic equation A 3 +A 2 + A + 21=0; but the 
latent roots are + 1 ± V— 6 and — 3 ; so the system is unstable. 

20/10. Another test, related to Nyquist's, states that a linear 
system is stable if, and only if, the polynomial 

l n + mj"- 1 + m 2 X n ~ 2 +--•+«■ 
changes in amplitude by nn when A, a complex variable 
(A = a + hi where i = V— 1), goes from - t oo to + t co along 
the fr-axis in the complex A-plane. 

Nyquist's criterion of stability is widely used in the theory 
of electric circuits and of servo-mechanisms. It, however, uses 
data obtained from the response of the system to persistent 
harmonic disturbance. Such disturbance renders the system 
non-absolute and is therefore based on an approach different from 
ours. 

20/11. Some further examples will illustrate various facts 
relating to stability. 

Example 1 : If a matrix [a] of order n x n has latent roots 
A l9 . . . , A n , then the matrix, written in partitioned form, 

! / 



of order 2n x 2/?, where / is the unit matrix, has latent roots 

± VI7, . . . , ± a/A„. It follows that the system 

d 2 x- 

— ! = di 1 x 1 + a i2 x 2 + . . . + ainXn (« = 1, . . . , «) 

of common physical occurrence, must be unstable. 

Example 2 : The diagonal terms an represent the intrinsic 
stabilities of the variables ; for if all variables other than xi are 
held constant, the linear system's i-th. equation becomes 
dxi/dt — auxi + c, 
219 



20/11 DESIGN FOR A BRAIN 

where c is a constant, showing that under these conditions Xi 
will converge to — c/au if an be negative, and will diverge without 
limit if an be positive. 

If the diagonal terms an are much larger in absolute magnitude 
than the others, the roots tend to the values of an. It follows 
that if the diagonal terms take extreme values they determine 
the stability. 

Example 3 : If the terms aij in the first n — 1 rows (or columns) 
are given, the remaining n terms can be adjusted to make the 
latent roots take any assigned values. 

Example 4 : The matrix of the homeostat equations of S, 19/11 
is 

1 



a^h a 12 h a 13 h a li h 
a 9 ,h a 99 h a 9 Ji a 9 Ji 



24' 



-J 



h a»Ji a^Ji anji 



33' 



-J 



_a il h a i9 h a i3 h a^Ji 



-1 



If j = o, the system must be unstable (by Example 1 above). 
If the matrix has latent roots fi v . . . , /u 8 , and if A l5 . . . , A 4 
are the latent roots of the matrix [a%jh] 9 and if j ^ 0, then the 
A's and ^'s are related by X p = jbt 2 q -f- jju q . As ; — > oo the 8-variable 
and the 4-variable systems are stable or unstable together. 

Example 5 : In a stable system, fixing a variable may make 
the system of the remainder unstable. For instance, the system 
with matrix 

6 5 - 10" 

- 4 - 3 - 1 
4 2 - 6 . 

is stable. But if the third variable is fixed, the system of the 
first two variables has matrix 



L-4 -3J 



and is unstable. 

Example 6 : Making one variable more stable intrinsically 

220 



STABILITY 



20/12 



(Example 2 of this section) may make the whole unstable. For 
instance, the system with matrix 



is stable. But if a lx becomes more negative, the system becomes 
unstable when a xl becomes more negative than — 4 J. 
Example 7 : In the n x n matrix 



a 



c i d 

in partitioned form, [a] is of order k X k. If the k diagonal 
elements an become much larger in absolute value than the rest, 
the latent roots of the matrix tend to the k values an and the 
n — k latent roots of [d]. Thus the matrix, corresponding to [d], 

1 
.1 
has latent roots -- 1-5 i l-658«, and the matrix 

— 1 2 0" 
100-1 2 

— 3 1—3 

— 1 1 2. 

has latent roots -- 101-39, — 98-62, and + 1-506 ± 1-720*. 

Corollary : If system [d] is unstable but the whole 4-variable 
system is stable, then making x x and x 2 more stable intrinsically 
will eventually make the whole unstable. 

Example 8 : The holistic nature of stability is well shown by 
the system with matrix 

— 3 — 2 2' 
-6-5 6 

— 5 2 — 4_ 
in which each variable individually, and every pair, is stable ; 
yet the whole is unstable. 



100 

— 2 



2 



The probability of stability 

20/12. The probability that a system should be stable can be 
made precise by the point of view of S. 14/16. We consider 

221 



20/12 DESIGN FOR A BRAIN 

an ensemble of absolute systems 

da%/dt =fi(x lt . . . , x n ; ol v . .-.) (i = 1, . . . , n) 

with parameters oy, such that each combination of a-values gives 
an absolute system. We nominate a point Q in phase-space, and 
then define the ' probability of stability at Q ' as the proportion 
of a-combinations (drawn as samples from known distributions) 
that give both (1) a resting state at Q, and (2) stable equilibrium 
at that point. The system's general ' probability of stability ' is 
the probability at Q averaged over all Q-points. As the proba- 
bility will usually be zero if Q is a point, we can consider instead 
the infinitesimal probability dp given when the point is increased 
to an infinitesimal volume dV. 

The question is fundamental to our point of view ; for, having 
decided that stability is necessary for homeostasis, we want to 
get a system of 10 10 nerve-cells and a complex environment 
stable by some method that does not demand the improbable. 
The question cannot be treated adequately without some quan- 
titative study. Unfortunately, the quantitative study involves 
mathematical difficulties of a high order. Non-linear systems 
cannot be treated generally but only individually. Here I shall 
deal only with the linear case. It is not implied that the nervous 
system is linear in its performance or that the answers found 
have any quantitative application to it. The position is simply 
that, knowing nothing of what to expect, we must collect what 
information we can so that we shall have at least some fixed 
points around which the argument can turn. 

The applicability of the concept of linearity is considerably 
widened by the theorem of S. 20/4. 

The problem may be stated as follows : A matrix of order 
n x n has elements which are real and are random samples from 
given distributions. Find the probability that all the latent 
roots have non-positive real parts. 

This problem seems to be still unsolved even in the special 
cases in which all the elements have the same distributions, 
selected to be simple, as the ' normal ' type e~ x , or the ' rect- 
angular ' type, constant between — a and -f- a. Nevertheless, 
some answer is desirable, so the ' rectangular ' distribution (integers 
evenly distributed between — 9 and + 9) was tested empirically. 
Matrices were formed from Fisher and Yates' Table of Random 

222 



STABILITY 



20/12 



Numbers, and each matrix was then tested for stability by Hurwitz' 
rule (S. 20/8 and S. 20/9). Thus a typical 3x3 matrix was 

— 1 - 3 -8' 

- 5 4—2 
_ 4 _ 4 _ 9^ 

In this case the second determinant is — 86, so it need not be 
tested further as it is unstable by S. 20/9. The testing becomes 
very time-consuming when the matrices exceed 3x3, for the time 
taken increases approximately as /i 5 . The results are summarised 
in Table 20/12/1. 



Order of 
matrix 


Number 
tested 


Number 
found 
stable 


Per cent 
stable 


2x2 
3x3 

4x4 


320 
100 
100 


77 
12 

1 


24 
12 

1 



Table 20/12/1. 



The main feature is the rapidity with which the probability 
tends to zero. The figures given arc compatible (x 2 = 4-53, 
P = 0-10) with the hypothesis that the probability for a matrix 
of order n x n is l/2 n . That this may be the Correct expression 
for this particular case is suggested partly by the fact that it 
may be proved so when n = 1 and n = 2, and partly by the 
fact that, for stability, the matrix has to pass all of n tests. 
And in fact about a half of the matrices failed at each test. 
If the signs of the determinants in Hurwitz' test are statistically 
independent, then l/2 n would be the probability. 

In these tests, the intrinsic stabilities of the variables, as 
judged by the signs of the terms in the main diagonal, were 
equally likely to be stable or unstable. An interesting variation, 
therefore, is to consider the case where the variables are all 
intrinsically stable (all terms in the main diagonal distributed 
uniformly between and — 9). 

The effect is to increase their probability of stability. Thus 
when n is 1 the probability is 1 (instead of J) ; and when n is 

223 



20/12 



DESIGN FOR A BRAIN 



2 the probability is 3/4 (instead of 1/4). Some empirical tests 
gave the results of Table 20/12/2. 



Order of 
matrix 


Number 
tested 


Number 
found 
stable 


Per cent 
stable 


2x2 
3X3 


120 

100 


87 
55 


72 
55 



Table 20/12/2. 

The probability is higher, but it still falls as n is increased. 

A similar series of tests was made with the homeostat. Units 
were allowed to interact with settings determined by the uni- 
selectors, and the percentage of stable combinations found when 
the number of units was two; the percentage was then found 
for the same general conditions except that three units interacted ; 

ioo 



50 




2 3 

number of variables 
Figure 20/12/1. 

and then four. The general conditions were then changed and 
a new triple of percentages found. And this was repeated six 
times altogether. As the general conditions sometimes encour- 
aged, sometimes discouraged, stability, some of the triples were 
all high, some all low ; but in every case the per cent stable fell 
as the number of interacting units was increased. The results 
are given in Figure 20/12/1. 

224 



STABILITY 20/12 

These results prove little ; but they suggest that the proba- 
bility of stability is small in large systems assembled at random. 
It is suggested, therefore, that large systems should be assumed 
unstable unless evidence to the contrary can be produced. 



References 

Ashby, W. Ross. The effect of controls on stability. Nature, 155, 242 ; 

1945. 
Idem. Interrelations between stabilities of parts within a whole dynamic 

system. Journal of comparative a?id jihysiological Psychology, 40, 1 ; 

1947. 
Idem. The stability of a randomly assembled nerve-network. Electro- 
encephalography and clinical neurophysiology, 2, 471 ; 1950. 
Frazer, R. A., and Duncan, W. J. On the criteria for the stability of small 

motions. Proceedings of the Royal Society, A, 124, 642 ; 1929. 
Hurwitz, A. t)ber die Bedingungen, unter welchen eine Gleichung nur 

Wurzeln mit negativen reellen Teilen besitzt. Mathematische Annalen, 

46, 273 ; 1895. 
Nyquist, H. Regeneration theory. Bell System technical Journal, 11, 126 ; 

1932. 



225 



CHAPTER 21 

Parameters 



21/1. With canonical equations 

— * = f i (x 1 , . . . , x n ) {i = 1, . . . , ri), 

the form of the field is determined by the functional forms ft 
regarded as functions of x v . . . , x n . If parameters a v a 2 , . . . 
are taken into consideration, the system will be specified by 
equations 

-j— = Ji\&ii . . . , x n ; fit 1? & 2 , • • •/ [t = x 9 , m , 9 71). 

If the parameters are constant, the #'s continue to form an absolute 
system. If the a's can take m combinations of values, then the 
oj's form m different absolute systems, and will show m different 
fields. If a parameter can change continuously (in value, not in 
time), no limit can be put to the number of different fields which 
can arise. 

If a parameter affects only certain variables directly, it will 
appear only in the corresponding /'s. Thus, if it affects only 
x x directly, so that the diagram of immediate effects is 

(X * X -i ^ Xn) 

then a will appear only in f x : 

dxj&t =f 1 (x 1 , x 2 ; a) 
dxjdt =f 2 {x lf x 2 ). 

But it will in general appear in all the F's of the integrals (S. 19/10). 
The subject is developed further in Chapter 24. 

Change of parameters can represent every alteration which can 
be made on an absolute system, and therefore on any physical 
or biological ' machine '. It includes every possibility of experi- 
mental interference. Thus if a set of variables that are joined 
to form the system x = f(x) are changed in their relations so 
that they form the system x = <j>(x), then the change can equally 

226 



PARAMETERS 21/2 

well be represented as a change in the single system x = ip(x ; a). 
For if a can take two values, 1 and 2 say, and if 

f(x) =yj(a:; 1) 
(j)(x) = ip(x ; 2) 
then the two representations are identical. 

As example of its method, the action of S. 8/10, where the two 
front magnets of the homeostat were joined by a light glass fibre 
and so forced to move from side to side together, will be shown 
so that the joining and releasing are equivalent in the canonical 
equations to a single parameter taking one of two values. 

Suppose that units x lt x 2 and x 3 were used, and that the 
magnets of 1 and 2 were joined. Before joining, the equations 
were (S. 19/11) 

dxjdt = a 11 x 1 + a 12 x 2 + a 13 x 3 ^\ 

dx 2 /dt = a 21 X ± -f- «22^2 + a 22 X 3 f 

dxjdt = a 31 x ± + a 32 x 2 + a 33 x 3 ) 

After joining, x 2 can be ignored as a variable since x x and x 2 are 
effectively only a single variable. But x 2 s output still affects the 
others, and its force still acts on the fibre. The equations there- 
fore become 

dxjdt = (a ±1 + a 12 + a 21 + a 22 )x x + {a 13 + a^)^ 
dxjdt = (a 31 + 032)^ + a 32 x 3 

It is easy to verify that if the full equations, including the parameter 

b t were : 

dxjdt = {a lx + b(a 12 + a 21 + a 22 )}x 1 + (1 - b)a 12 x 2 

+ (« is + ^23)^3 
dxjdt = a 21 x x + a 22 x 2 + o 23 x 3 

dxjdt = (a 31 + ^32)^! + (1 — b)a 32 x 2 + a 33 x 3 _ 

then the joining and releasing are identical in their effects with 
giving b the values 1 and respectively. (These equations are 
sufficient but not, of course, necessary.) 

21/2. A variable x^ behaves as a ' null- function ' if it has the 
following properties, which are easily shown to be necessary and 
sufficient for each other : 

(1) As a function of the time, it remains at its initial value x% 

(2) In the canonical equations, fk(x lt . . . , x n ) is identically 

zero. 

227 



21/3 DESIGN FOR A BRAIN 

(3) In the group equations, F k (x®, . . . , a?J ; t) = x^. 
(Some region of the phase-space is assumed given.) 
Since we usually consider absolute systems, we shall usually 
require the parameters to be held constant. Since null-functions 
also remain constant, the properties of the two will often be 
similar. (A fundamental distinction by definition is that para- 
meters are outside, while null-functions may be inside, the given 
system.) 

21/3. In an absolute system, the variables other than the step- 
and null-functions will be referred to as main variables. 

21/4. Theorem : In an absolute system, the system of the main- 
variables forms an absolute subsystem provided no step-function 
changes from its initial value. 

Suppose x l9 . . . , Xjt are null- and step-functions and the main- 
variables are Xk+u . . . , x n . The canonical equations of the 
whole system are 

dxjdt = 



dxjc/dt — 
dxk+i/dt = fk+i(x v . . . , x k , x k+ i, . . . , x n ) 



dXn/dt =f n (x v . . . , X*, X k+1 , . . . , X n ) 

The first k equations can be integrated at once to give x x = x\, 
. . ., Xk = x Q k . Substituting these in the remaining equations 
we get : 

dx k+ i/dt = fk+i{x\, . . ., x% x k +i f . . ., x n T\ 



ClX n /dt — Jn\pC\i • • • 5 #jfc> Xk+1, • • •» x n ) J 

The terms x^, . . ., x^ are now constants, not effectively functions 
of t at all. The equations are in canonical form, so the system is 
absolute over any interval not containing a change in a?J, . . . , x^. 
Usually the selection of variables to form an absolute system 
is rigorously determined by the real, natural relationships existing 
in the real ' machine ', and the observer has no power to alter them 
without making alterations in the ' machine ' itself. The theorem, 
however, shows that without affecting the absoluteness we may take 

228 



PARAMETERS 21/6 

null-functions into the system or remove them from it as we 
please. 

It also follows that the statements : ' parameter a was held con- 
stant at a \ and c the system was re-defined to include a, which, 
as a null-function, remained at its initial value of a ' are merely 
two ways of describing the same facts. 

21/5. The fact that the field is changed by a change of parameter 
implies that the stabilities of the lines of behaviour are changed. 
For instance, consider the system 

dx/dt = — x -f ay, dy/dt = x — y -f 1 
where x and y have been used for simplicity instead of x ± and x 2 . 
When a — 0, 1, and 2 respectively, the system has the three 
fields shown in Figure 21/5/1. 





Figure 21/5/1 : Three fields of x and y when a has the values (left to 
right) 0, 1, and 2. 

When a = there is a stable resting state at a? = 0, y = 1 ; 

when a = 1 there is no resting state ; 

when a = 2 there is an unstable resting state at x = — 2, 

y = -l. 
The system has as many fields as there are values to a. 



21/6. The simple physical act of joining two machines has, of 
course, a counterpart in the equations, shown more simply in the 
canonical than in the group equations. 

One could, of course, simply write down equations in all the 
variables and then simply let some parameter a have one value 
when the parts are joined and another when they are separated. 
This method, however, gives no insight into the real events in 
' joining ' two systems. A better method is to equate para- 
meters in one system to variables in the other. When this is 

229 q 



21/7 DESIGN FOR A BRAIN 

done, the second dominates the first. If parameters in each are 
equated to variables in the other, then a two-way interaction 
occurs. For instance, suppose we start with the 2-variable 
system 

dx/dt = fJx, y; a)\ , .. „ . . . _ . _ 

, ,, _ // \ fand the 1 -variable system dz/dt = 6(z; b) 
ay /at — j 2 (x, y) j 

then the diagram of immediate effects is 

a— > x+±y b—> z 

If we put a = z, the new system has the equations 

dx/dt =f 1 {x, y; z)\ 

dy/dt =f 2 {x, y) > 

dz/dt = cf>(z ; b) J 
and the diagram of immediate effects becomes 

b — > z — ► x ^=t y. 
If a further join is made by putting b = y, the equations become 

dx/dt —fiix, y; z) 
dy/dt =f 2 (x, y) 
dz/dt = <j>(z ; y) 
and the diagram of immediate effects becomes 



In this method each linkage uses up one parameter. This is 
reasonable ; for the parameter used by the other system might 
have been used by the experimenter for arbitrary control. So 
the method simply exchanges the experimenter for another 
system. 

This method of joining does no violence to each system's 
internal activities : these proceed as before except as modified by 
the actions coming in through the variables which were once 
parameters. 

21/7. The stabilities of separate systems do not define the 
stability of the system formed by joining them together. 

In the general case, when the/'s are unrestricted, this propo- 
sition is not easily given a meaning. But in the linear case (to 

230 



PARAMETERS 21/7 

which all continuous systems approximate, S. 20/4) the meaning is 
clear. Several examples will be given. 

Example 1 : Two systems may be stable if joined one way, and 
unstable if joined another. Consider the 1 -variable systems 
dx/dt = x + 2p x -f Vz an d dy/dt = — 2r — 3y. If they are 
joined by putting r = x, p x = y, the system becomes 

dx/dt = x -f 2y + p, 

dy/dt = — 2x — 3y 

The latent roots of its matrix are — 1, — 1 ; so it is stable. But 

if they are joined by r - x, p 2 = y, the roots become + 0-414 

and — 2-414 ; and it is unstable. 

Example 2 : Several systems, all stable, may be unstable when 
joined. Join the three systems 

dx/dt = — x — 2q — 2r 
dy/dt = — 2p — y -f- r 
dz/dt = p -f a — z 
all of which are stable, by putting p = x, q = y, r = z. The 
resulting system has latent roots +1, — 2, — 2. 

Example 3: Systems, each unstable, may be joined to form a 
stable whole. Join the 2-variable system 

dx/dt = Sx — Sy — Sp 

dy/dt = 3x — 9y — 8p^ 
which is unstable, to dz/dt = 21 g -j- 3r -J- 3^, which is also 
unstable, by putting q — x, r = y, p = z. The whole is stable. 
Example 4 : If a system 
dxi/dt =fi{x v . . . , x n ; a l9 . . .) (i = 1, . . . , n) 

is joined to another system, of ?/'s, by equating various a's and i/'s, 
then the resting states that were once given by certain com- 
binations of x and a will still occur, so far as the ^-system is 
concerned, when the ?/'s take the values the a's had before. The 
zeros of the/'s are thus invariant for the operations of joining and 
separating. 



231 



CHAPTER 22 

Step-Functions 

22/1. A variable behaves as a step-function over some given 
period of observation if it changes value at only a finite number of 
discrete instants, at which it changes value instantaneously. 
The term ' step-function ' will also be used, for convenience, to 
refer to any physical part whose behaviour is typically of this 
form. 



22/2. An example of a step-function in a system will be given 
to establish the main properties. 

Suppose a mass m hangs downwards suspended on a massless 
strand of elastic. If the elastic is stretched too far it will break 
and the mass will fall. Let the elastic pull with a force of k 
dynes for each centimetre increase from its unstretched length, 
and, for simplicity, assume that it exerts an opposite force when 
compressed. Let x, the position of the mass, be measured verti- 
cally downwards, taking as zero the position of the elastic when 
there is no mass. 

If the mass is started from a position vertically above or below 
the point of rest, the movement will be given by the equation 



/ dx\ 
{ m dt) 



where g is the acceleration due to gravity. This equation is not 
in canonical form, but may be made so by writing x = x Xi 
dx/dt = x 2 , when it becomes 



(2) 




232 



STEP-FUNCTIONS 



22/2 



If the elastic breaks, k becomes 0, and the equations become 

dx 1 



dt 
dx 2 
~dt 



(3) 



Assume that the elastic breaks if it is pulled longer than X. 

The events may be viewed in two ways, which are equivalent. 

We may treat the change of k as a change of parameter to the 
2-variable system x v x 2 , changing their equations from (2) 
above to (3) (S. 21/1). The field of the 2-variable system will 
change from A to B in Figure 22/2/1, where the dotted line at X 





A B 

Figure 22/2/1 : Two fields of the system {x x and x 2 ) of S. 22/2. 
unbroken elastic the system behaves as A, with broken as B. 
the strand is stretched to position X it breaks. 



With 
When 



shows that the field to its right may not be used (for at X the 
elastic will break). 

Equivalent to this is the view which treats them as a 3- 
variable system : sc l9 x 2 , an d k. This system is absolute, and has 
one field, shown in Figure 22/2/2. 

In this form, the step-function must be brought into the 
canonical equations. A possible form is : 

dk (K K 
dt = q [-2 +2 

where K is the initial value of the variable k, and q is large and 
positive. As q— ■> oo, the behaviour of k tends to the step- 
function form. 

Another method is to use Dirac's ^-function, defined by S(u) = 
if u ?±0, while if u = 0, d(u) tends to infinity in such a way that 

rco 

I d(u)du = 1. 

J —00 

233 



+ - tanh {q(X - x,)} - k 



(4) 



22/2 



DESIGN FOR A BRAIN 



Then if du/dt = <5{</>(w, v, . . .)}, du/dt will be usually zero ; but 
if the changes of w, v , . . . take </> through zero, then d(u) becomes 
momentarily infinite and n will change by a finite jump. These 




Figure 22/2/2 : Field of the 3-variable system. 

representations are of little practical use, but they are important 
theoretically in showing that a step-function can be represented 
in the canonical equations. 

22/3. In an absolute system, a step-function will change value 
if, and only if, the system arrives at certain states : the critical. 
In Figure 22/2/2, for instance, all the points in the plane k = K 
and to the right of the line x x = X are critical states for the step- 
f unction k when it has the initial value K. 

The critical states may, of course, be distributed arbitrarily. 
More commonly, however, the distribution is continuous. In this 
case there will be a critical surface 

<f>(k t X{, . . . , X n ) = 
which, given k, divides the critical from the non-critical states. 
In Figure 22/2/2, for instance, the surface intersects the plane 
k = K at the line x 1 = X. (The plane k = is not intersected by 
it, for there are no states in this system whose occurrence will 
result in k changing from 0.) 

Commonly <£ is a function of only a few of the variables of the 

231 



STEP-FUNCTIONS 22/5 

system. Thus, whether a Post Office-type relay opens or shuts 
depends only on the two variables : the current in the coil, and 
whether the relay is already open or shut. 

Such relays and critical states occur in the homeostat. When 
two, three or four units are in use, the critical surfaces will form a 
square, cube, or tesseract respectively in the phase-space around 
the origin. The critical states will fill the space outside this sur- 
face. As there is some ' backlash ' in the relays, the critical 
surfaces for opening are not identical with those for closing. 



Systems with multiple fields 

22/4. If, in the previous example, someone unknown to us were 
sometimes to break and sometimes to replace the elastic, and if 
we were to test the behaviour of the system x v x 2 over a prolonged 
time including many such actions, we would find that the system 
was often absolute with a field like A of Figure 22/2/1, and often 
absolute with a field like B ; and that from time to time the field 
changed suddenly from the one form to the other. 

Such a system could be said without ambiguity to have two 
fields. Similarly, if parameters capable of taking r combinations 
of values were subject to intermittent change by some other, 
unobserved system, a system might be found to have r fields. 

22/5. The argument can, however, be reversed : if we find that 
a subsystem has r fields we can deduce, subject to certain restric- 
tions, that the other variables must include step-functions. 

Theorem : If, within an absolute system x v . . . , x n , x p , . . . , x s , 
the subsystem x l9 . . . , x n is absolute within each of r fields 
(which persist for a finite time and interchange instantaneously) 
and is not independent of x Pi . . . , x s ; then one or more of 
Xp, . . . , x s must be step-functions. 

Consider the whole system first while one field persists. Take 
a generic initial state x\, . . . , x% x^, . . . , x° s and allow time t x 
to elapse ; suppose the representative point moves to a?i, . . . , x n , 

x ' v x s , where each x' is not necessarily different from x°. 

Let further time t 2 elapse, the point moving on to x'u • • • > #n> 
x'p, . . . , x". Now consider the line of behaviour that follows 
the initial state x[, . . . , x n , x° p , x q , . . . , x' s , differing from the 

235 



22/5 DESIGN FOR A BRAIN 

second point only in the value of x p : as the subsystem is absolute, 
an interval t 2 will bring its variables again to x\, • • • , ®»» i-e. these 
variables' behaviours are the same on the two lines. Now x p 
either is, or is not, equal to x° r If unequal, then by definition 
(S. 14/3) x 19 . . . , x n is independent of x v . So the behaviour 
of x 19 . . . , x n over t 2 will show either that x' v = x® (i.e. that 

x p did not change over t ± ) or that x x x n is independent of 

x p . Similar tests with the other variables of the set x v , . . . , x s 
will enable them to be divided into two classes : (1) those that 
remained constant over t v and (2) those of which the subsystem 
X l9 . . . , X n is independent. By hypothesis, class (2) may not 
include all of x p , . . . , x g ; so class (1) is not void. 

When a field of x v . . . , x n changes, some parameter to this 
system must have changed value. As x lt . . . , x n , x p , . . . , x 8 
is isolated, the ' parameter ' can be none other than one or more of 
x Pi . . . , x s . As the field has changed, the parameter cannot be 
in class (2). At the change of field, therefore, at least one of 
those in class (1) changed value. So class (1), and therefore the 
set x p , . . . , Xs, contains at least one step-function. 

Reference 

Ashby, W. Ross. Principles of the self-organising dynamic system. Journal 
of general Psychology, 37, 125 ; 1947. 



236 



CHAPTER 23 

The Ultrastable System 



23/1. The definition and description already given in S. 8/6 and 
7 have established the elementary properties of the ultrastable 
system. A restatement in mathematical form, however, has the 
advantage of rendering a misunderstanding less likely, and of 
providing a base for quantitative studies. 

If a system is ultrastable, it is composed of main variables Xi 
and of step-functions at, so that the whole is absolute : 

-£ =fi(x; a) (i = 1, . . . , n) 

d ^ = gi(x; a) (i = l, 2, . . .) 

The functions gi must be given some form like that of S. 22/2. 
The system is started with the representative point within the 
critical surface cf)(x) = 0, contact with which makes the step- 
functions change value. When they change, the new values are 
to be random samples from some distribution, assumed given. 

Thus in the homeostat, the equations of the main variables are 
(S. 19/11) : 

dx' 

-± = a il x 1 + a i2 x 2 + a i3 x 3 + a ti x± (i = 1, 2, 3, 4) 

The a's are step-functions, coming from a distribution of ' rect- 
angular ' form, lying evenly between — 1 and -f- 1. The critical 

surfaces of the a's are specified approximately by | x \ ± - = 0. 

4 

Each individual step-function a^ depends only on whether Xj 
crosses the critical surface. 

As the a's change discontinuously, an analytic integration of 
the differential equations is not, so far as I am aware, possible. 
But the equations, the description, and the schedule of the 
uniselector-wirings (the random samples) define uniquely the 
behaviour of the x's and the a's. So the behaviour could be 

237 



23/2 DESIGN FOR A BRAIN 

computed to any degree of accuracy by a numerical method. The 
proof given in Chapter 8, though verbal, is adequate to establish 
the elementary properties of the system. A rigorous statement 
and proof would add little of real value. 

23/2. How many trials will be necessary, on the average, for a 
terminal field to be found ? If an ultrastable system has a 
probability p that a new field of the main variables will be stable, 
and if the fields' probabilities are independent, then the number 
of fields occurring (including the terminal) will be, on the average, 

i/ P . 

For at the first field, a proportion p will be terminal, and 
q (= 1 — p) will not. Of the latter, at the second field, the pro- 
portion p will be terminal and q not ; so the total proportion stable 
at the second field will be pq, and the number still unstable q 2 . 
Similarly the proportion becoming terminal at the w-th field will 
be pq u ~ A . So the average number of trials made will be 
p -f 2pq + 3pq 2 + . . . + upq u ~ x + . . . _ 1 
V + V9. + Vf + • • • + Vt~ x + • • • ~ V 

23/3. In an ultrastable system, a field may be terminal and yet 
show little resemblance to the ' normal ' equilibrium which is 
necessary if the system is to show, after each of a variety of dis- 
placements, a return to the resting state. A field, for instance, 
might have a resting state at which only a single line of behaviour 
terminated : if the representative point were on that line the field 
would be terminal ; but hardly any displacement would be 
followed by a return to the resting state. 

It can, however, be shown that if a proportion of the fields 
evoked by the step-function changes are of this or similar type, 
then the terminal fields will contain them in smaller proportion. 
For, given a field and a closed critical surface, let k ± be the pro- 
portion of lines of behaviour crossing the boundary which are 
stable. Thus in Figure 8/7/1, in I k x = 0, in II h t = 0, in III 
/c 1 = i approximately, and in IV k ± = 1. To count the lines, 
the boundary surface could be divided into portions of equal 
area, small enough so that stable and unstable lines do not pass 
through the same area. Then if we assume that in any field the 
representative point is equally likely to start at any of the small 
areas, a field's chance of being terminal is proportional to k v 

238 



THE ULTRASTABLE SYSTEM 



23/4 



It follows that if the changes of step-functions evoke fields whose 
values of k 1 are distributed so that the probability of a field having 
a A^-value between k x and k t + dk x is ^(k-^dk^ then in the terminal 
fields the probablity is 

k 1 y)(k 1 )dk 1 



a) 



k 1 y)(k 1 )dk 1 



Figure 23/3/1 shows a possible distribution of values of k 1 in 



> 




u 




z 




LU 




13 




O 




LU 




a. 




u_ 





Figure 23/3/1 : Solid line : a distribution y^i) 5 broken line : 
the corresponding distribution k^kj. 

the original fields (solid line), and how k ± would then be dis- 
tributed in the terminal fields (broken line). The shift towards 
the higher values of k x is clear. 

Fields with a low value of k v unsatisfactory for adaptation, 
tend therefore not to be terminal. 



23/4. It was noticed in S. 13/4 that fields like A and B of Figure 
13/4/1, though terminal, are defective in their persistence after 
small random disturbances. This idea may be given more 
precision. 

Assume that the small random disturbances cause displace- 
ments which have some definite probability distribution, Gaussian 
say, so that if applied to the representative point when it is at 
some definite position in the field, there is a definite probability 
k 2 that a random displacement will not carry the point beyond 
the critical surface. Assume the representative point is always 
at the resting state or resting cycle. Then any terminal field has 
a unique value for k 2 . If the field contains a single resting state, 
k 2 for that field is the probability, when the representative point 

239 



23/4 DESIGN FOR A BRAIN 

is at the resting state, that the application of a single random dis- 
turbance will not take the representative point beyond the critical 
surface. If the field has a resting cycle, k 2 is the average of the 
values when the representative point is on the many portions of 
the cycle, the value for each portion being weighted according 
to the time spent by the representative point in that portion. For 
more complex fields, k 2 could be defined, but a more detailed 
study is not necessary here. 

Suppose that the ultrastable system, when the step-functions 
undergo random changes, yields terminal fields whose values of k 2 
are distributed so that the proportion falling between k 2 and 
k 2 + dk 2 is cf>(k 2 )dk 2 . If to such fields, with k 2 lying between such 
limits, we apply one random disturbance, a proportion k 2 will 
not be changed ; but the proportion 1 — k 2 will be changed, and 
will be replaced by new terminal fields ; their values of k 2 will be 
distributed again as <j)(k 2 )dk 2 , and this distribution will be added 
to that of the unchanged fields. In this way it is easy to show 
that the final distribution X{k 2 ) equals 

where A is a constant. 

Examination of the form of the distribution k{k 2 ) shows that 
it is cf>{k 2 ) heavily weighted in favour of the values of k 2 near 1. 
Such fields can only be those with the resting state or cycle near 
the centre of the region. So the result confirms the common- 
sense argument of S. 13/4. It will be noticed that the deduction 
is independent of the particular form of the distribution of 
disturbances. 

References 

Asiiby, W. Ross. The physical origin of adaptation by trial and error. 

Journal of general Psychology, 32, 13 ; 1945. 
Idem. The nervous system as physical machine : with special reference to 

the origin of adaptive behaviour. Mind, 56, 1 ; 1947. 



240 



CHAPTER 24 

Constancy and Independence 

24/1. The relation of variable to variable has been treated by 
observing the behaviour of the whole system. But what of their 
effects on one another ? Thus, if a variable changes in value, can 
we distribute the cause of this change among the other variables ? 
In general, it is not possible to divide the effect into parts, 
with so much caused by this variable and so much caused by that. 
Only when there are special simplicities is such a division possible. 
In general, the change of a variable results from the activity of 
the whole system, and cannot be subdivided quantitatively. 
Thus, if dx/dt = sin x + xe y , and x = \ and y = 2, then in the 
next 0-01 unit of time x will increase by 0-042, but this quantity 
cannot be divided into two parts, one due to x and one to y. 

24/2. But a relationship which can be treated in detail is that of 
' independence '. By the principle of S. 2/8 it must be defined in 
terms of observable behaviour. 

Given an absolute system and two lines of behaviour from two 
initial states which differ only in their values of x® (the difference 
being A#°), the variable x k is independent of Xj if x k 's behaviour is 
identical on the two lines. Analytically, x k is independent of Xj 
in the conditions given if 

F k (4, . . . , x% . . . ; t) = F k (xl . . . , ^ + AflJ, . . . ; t) (1) 
as a function of t. In other words, x k is independent of Xj 
if XfcS behaviour is invariant whemthe initial state is changed 
by A4 

This narrow definition provides the basis for further develop- 
ment. In practical application, the identity (1) may hold over all 
values of Aa?° (within some finite range, perhaps) ; and may also 
hold for all initial states of x k (within some finite range, perhaps). 
In such cases the test whether x k is independent of Xj is whether 

241 



24/3 DESIGN FOR A BRAIN 

r-Q Ffc(aj?, . . . , a?2 » /) = 0. (These relations and notations are 

collected in S. 24/19 for convenience in reference.) 
Example : In the system of S. 19/10 

X-^ I= X^ -j- XyZ ~i~ t 

x 2 = x% + 2t 
x 2 is independent of x v but x x is not independent of x 2 . 

24/3. We shall be interested chiefly in the independencies intro- 
duced when particular variables become constant : when they are 
part-functions, for instance. Such constancies are most naturally 
expressed in the canonical equations, for here are specified the 
properties of the parts before assembly (S. 19/19). We there- 
fore need a method of deducing the independence from the 
canonical equations, preferably without an explicit integration. 
Such a method is developed below in S. 24/3 to 10. (The method 
recently developed by Riguet, however, promises to be much 
better.) 

Given an absolute system 

-^ =f i (x 1 , . . . , x n ) (t = 1, . . - , n) . (1) 

it is required to find whether or not x^ is independent of Xj, some 
region of values being assumed. The region must not include 
changes of values of step-functions or of activations of part- 
functions ; for the derivatives required below may not exist, 
and the independencies may change. 

If the functions fi are expandable by Taylor's series around the 
point X®, . . . , x„, we may write their integrals symbolically 
(S. 19/27) as 

Fi(4, . . . , xl ; t) = <**x\ {i = 1, . . . , n) . (2) 

where X is the operator 

/K . . . , a£)£g + . . . +f n {xl . . . , fl©gjg. 

(The zero superscripts will now be dropped as unnecessary.) 

pi 

Expanding the exponential, and operating on (2) with =— , 
the test whether x k is independent of Xj becomes whether 

sf— <*- 1. *...•> • • (3) 

242 



CONSTANCY AND INDEPENDENCE 24/5 

By expanding ^— A': 

£- X'+'a:* = £& £- X*x t + X ~ X»x k . (4) 

OXj '—' OXj OXp OXj 

Applying the test (3), if the test for /j, = m gives 

5- X m X k = 

OXj 

then for /li = m -j- 1, by using (4) we need only see whether 

?%k**- ■ ■ ■ ^ 

24/4. We now add the hypothesis that the system is linear (S. 
19/27). The restriction is unimportant as no arguments are used 
elsewhere which depend on linearity or on non-linearity. Further, 
in the region near a resting state all systems tend to the linear 
form (S. 20/4), and this region has our main interest. 
Starting with ju = 1 the tests 24/3 (5) become 

OXj 

sfi a/, _ 



z 



dx p dxj 

~* — ' dx p dx a dxj 
etc. 



(i) 



These tests now use only the /'s, as required. They are both 
necessary and sufficient. They have been shown necessary ; and 
by merely retracing the argument they are found to be sufficient. 
Only the first n — 1 tests of (1) above are required, for products 
which contain more than n — 1 factors must include products 
already given, in the first n — 1 tests, as zero. 

The tests are, however, clumsy. The simplicity and directness 
can be improved by using the facts that we need distinguish only 
between zero and non-zero quantities, and that the sums of (1) 
above resemble the elements of matrix products. Sections 24/5-10 
develop this possibility. 

24/5. An R0- matrix has elements which can take only two 
values : R (non-zero) and (zero). The elements therefore 

243 



24/6 DESIGN FOR A BRAIN 

combine by the rules 

R + R = R, 0+0=0, # + 0=0 + ]? = #, 
R x R = R, 0x0=0, Rx0=0xR=0. 

A sum of such elements can therefore be zero in general only if 

each element is zero. 

24/6. In an 7?0-matrix of order n x n, the zeros are patterned 
if, given any zero not in the principal diagonal, we can separate 
the numbers 1, 2, . . . , n into two sets a and /? (neither being 
void) so that the minor left after suppressing columns a and rows /? 
is composed wholly of zeros which include the given zero. For 
example, the 720-matrix 



R R 

R 





R 
R 
R 

R 

has its zeros patterned. Selecting, for instance, 
the third row, we can make a = 1, 3, 4 and /? = 2. 
the minor 





the zero in 
This leaves 



. 
. 

where dots indicate eliminated elements ; the remaining elements 
are all zero, and they include the selected zero. The other zeros 
in the original matrix can all be treated similarly. 

24/7. Some necessary theorems will now be stated. Their proofs 
are simple and need not be given here. 
A matrix A is idempotent if A 2 = A. 
Theorem : If an 7?0-matrix has no zeros in the principal diagonal 
a necessary and sufficient condition that the zeros be patterned 
is that the matrix be idempotent. 



24/8. Theorem : If A is an i?0-matrix of order n X w, and I 
is the matrix with 72's in the principal diagonal and zeros elsewhere, 
then the matrix 

I + A + A 2 + . . . + A n ~ x 
is idempotent. 

244 



CONSTANCY AND INDEPENDENCE 24/10 

24/9. From the /'s of the canonical equations (24/3(1) ) form 
the differential matrix [f] by inserting, in the (kj)-th position 
(at the intersection of the A:-th row and the ;'-th column) an or 
R according as dfk/dxj is, or is not, zero (in the region of phase- 
space considered). Then the square, cube, etc., of [/] will contain 
in the (A^')-th position an element which is zero or non-zero as the 
second, third, etc., tests of 24/4(1) are or are not zero. If now 
these powers are summed, to S : 

« = m + [/? + • • • + 1/]- 1 , . . (i) 

a zero element in S at the (Jcj)-th position means that all the 
terms of the series were zero, and therefore that Xk is independent 

Of Xj. 

The same independence will make zero the element at the {kj)-t\\ 
position in the matrix whose (^)-th element is zero or non-zero 
as dFn/dJl is or is not zero. This integral matrix, [F], must 
therefore satisfy 

[*]=« (2) 

24/10. The restriction is now added that the behaviour of each 
variable xt is to depend on its own starting-point. (Physical 
systems not conforming to this restriction are, so far as I am 
aware, rare and peculiar.) The principal diagonal of [F] will 
then be found to have all its elements non-zero. In such a case, 
[F] is not altered if we add to it the matrix / of S. 24/8, and we may 
sum up as follows : 

If a dynamic system is specified by 

— l =f i {x 1 , . . . , x n ) (i = 1, . . . , n) 

and if [/] is an jRO-matrix where each (ftj)-th element is or R 

as ~- is or is not zero respectively (in some region within which 

CXj 

the nullity does not change), and if [F] is an .RO-matrix where 
each (kj)-th element is or J? as Xk is or is not independent of Xj 
respectively in the same region, and if each x's behaviour depends 
on its own starting point, then 

m = [/] + [/]« + • • ■ + [n n - 1 ■ ■ a) 

This equation gives the independencies when the differential 
matrix is given; for x k is or is not independent of x 5 as the 

245 r 



24/11 DESIGN FOR A BRAIN 

element in the k-th row and the j-th column of the integral 
matrix is or is not zero respectively. 

The advantage of equation (1) is that the differential matrix 
is often formed with ease (for only zero or non-zero values are 
required), and often the first multiplication shows that [/] 2 = [/]. 
When this is so, the integral matrix is at once proved to be equal 
to [/], and all the independencies are obtained at once. A 
further advantage is that the theory of partitioned matrices can 
often be used, with considerable economy of time. The next 
few sections provide some examples. 

24/11. In an absolute system the independencies cannot be 
assigned arbitrarily. 

By the theorem of S. 24/8, the integral matrix, being the sum 
of powers, is idempotent ; and therefore, by S. 24/7, has its zeros 
patterned. The independencies of an absolute system must 
always be subject to this restriction. 

What is really the same line of reasoning may be shown in an 
alternative form. The group property requires (S. 19/10) that 

F k {F x (x« ; /), F 2 (x» ;*),...;«'} = *W4 4 - • - I * + O; 
so if X/c is independent of Xj then x° will not appear effectively on 
the right-hand side, and it must therefore not appear effectively 
on the left. So if, say, F m ( . . . ; t) contains x°p then x ^ must not 
occur in F k ; so x k must be independent of x m as well. 

24/12. If the variables of an absolute system are divisible into 
two groups A and B, such that all the variables of A are inde- 
pendent of B, but not all those of B are independent of 4> then 
the subsystem A dominates the subsystem B. 

Theorem : The subsystem A is itself absolute. 

Write down the group equations of the A's : 

F A {F^; t), F 2 (x°; /),...; t'} = F A {x° v «& . . .; t + t'} 
where the subscript a refers to all the members of A in succession. 
Each F A is independent of a?jj, so, omitting the unnecessary 
symbols from each side both from the F's and from the x°'s, we get 

F A {F A {xl; t), . . .; t'} = F A {x° A ; t + t'} 
where the change of subscript means that only the members of 
A are now included. Inspection shows that these are the equa- 

246 



CONSTANCY AND INDEPENDENCE 24/14 

tions of a finite continuous group in the variables A. So the 
^4's form an absolute system. 

The fact of dominance may be shown in the integral matrix by 
finding that the deletion of columns A and rows B leaves only 
zeros ; but the deletion of columns B and rows A leaves some 
non-zero elements. (If the second operation also leaves only 
zeros, then the system really consists of two completely inde- 
pendent subsystems ; the whole system is 4 reducible '.) 

24/13. If A, B, and C are systems such that they together form 
one absolute system, and if A dominates B, and B dominates C, 
then A dominates C. 

On the information given, [F], in partitioned form, can be 
filled in but for two elements, shown as dots : 





A 


B 


C 


A 


~R 





. 


B 


R 


R 





C 


. 


R 


R 



It must be idempotent (S. 24/11). Trying the four possible 
combinations of R and for the two undefined elements, we find 
that there must be at the top right corner, and R at the bottom 
left. A therefore dominates C. 

The theorem illustrates again the importance of the concept of 
' absoluteness ' ; for without this assumption the theorem, 
obvious physically, cannot be proved (for lack of the group 
property). 

24/14. An account of the primary effects of part-functions on 
the independencies within an absolute system can now be given. 
The definition of a part-function x p implies that over finite 
regions of values of x l9 . . . , x n f p [oc l9 . . . , x n ) becomes zero. 
Within such a region, i.e. while not activated, the canonical 
equations include dx p /dt = 0, which can be integrated at once to 
x p = a£ ; so F p {x° ; t) = x° p ; and x p and F p are both constant. 
dF p /dxj is therefore zero for all values of j other than p. The 
effect of a part-function x p being inactive is therefore to make 
the whole of the p-th row of the differential and integral matrices 
zero (except for the element in the main diagonal, which remains 
an R). 

247 



24/14 



DESIGN FOR A BRAIN 



It will be recognised that [/] and [F], the differential and 
integral matrices, are the matrix equivalents of the diagrams of 



A. 



■^2 
A 




B 



—►3 

Figure 24/14/1. 



immediate and ultimate effects respectively. Thus the diagram 
of immediate effects A in Figure 24/14/1 yields the diagram of 
ultimate effects B. For the system, [/] is 



R 








o" 


R 


R 


R 











R 


R 





R 





r] 



assuming (S. 24/10) that the terms in the main diagonal are all R. 
Then 



in 



R 





o" 


R R 


R 


R 


R 


R 


R 


R R 


R 


R. 



[f? = 



R 








o" 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R. 



and the sum /+[/]+ l/T + L/T gives 



R 








" 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R. 



This, by S. 24/10, is the integral matrix. If it is compared directly 
with B of the Figure, the agreement will be found complete. 
Thus, it may be verified in both that x x dominates the system of 



it/05 ^3 anu & a. 



The rule of S. 14/10 for the formation of the diagram of ulti- 
mate effects when a variable is an inactive part-function can now 
be proved. For the effect on the differential matrix of a part- 
function xi being inactive is to make all the elements in the i-th. 
row zero, except the element in the main diagonal. Exactly 
the same change is caused in the differential matrix if we remove 
those arrows whose heads are at a&i. After these two changes 

248 



CONSTANCY AND INDEPENDENCE 24/15 

the correspondence continues as before. Thus, A of Figure 

14/10/1 has 

[/] = R R and I**] 



R 


R 





R~ 





R 





R 





R 


R 





R 





R 


R. 



R 


R 


R 


R~ 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R 


R. 



And a? 3 is not independent of x v But if cc 2 becomes inactive, 



[/] = 



[F] = 



R R R 

R 

R R 

R R R_ 

and x 3 is now independent of a^. 

The other diagrams, B, D and E, may be verified similarly. 



R 


# 


R 


R~ 





R 











R 


R 





# 


R 


R 


R. 



24/15. We can now investigate the problem of S. 14/8 : the 
separation of parts in a dynamic whole. 

Theorem : If the variables of an absolute system are divisible 
into three sets, A, B, and C such that no f A contains any of the 
set a&c, and no fc contains any of x A , i.e. so that the diagram of 
immediate effects is A ^ B ~^_ C, and if variables Xb remain 
constant, then A is independent of C, and vice versa. 

If all variables of set B are constant, the differential matrix, 
in partitioned form, will be 





A 


B 


C 


A 


~R 


R 


o" 


B 





R 





C 


_0 


R 


R. 



It is idempotent, so this matrix is also the integral matrix. As 
the elements at the top right and bottom left corners are zero, 
A and C are independent of each other. 

On the other hand, without further restrictions the constancy 
is not necessary. Thus, suppose that A and C are independent 
and that the differential matrix is 

R R (T 
P R Q 
.0 R R^ 

where P and Q are to be determined. For the integral matrix 
to have zeros in the top right and bottom left corners, it is easily 

249 



24/15 DESIGN FOR A BRAIN 

found that P and Q must both be zero. So df B /dx A and df B /dx c 
must be zero over the region. This can be achieved in several 
ways without fn being zero, i.e. without Xb being constant. Two 
examples will be given. 

(1) If fn is a constant, then Xb will increase uniformly, i.e. will 
not be constant, but Xa and xc will still be independent. Without 
a fourth variable, the linear change is the most which x B can make 
if the system is to remain absolute. 

(2) If fn is a function of other variables not yet mentioned, y 
is not restricted to a constant rate of change. Thus if there is a 
variable u which dominates y we could have a system 



dx 
~dt 



x + y 



du _ 
dt~ 3 



dy 



= sin u 



dt 
dz 

dt= y+Z j 
which is clearly absolute. Its solution is : 



x = (x Q + y° + 



sin u° + — cos u )e l — y° 



cos IT 
3 



10 



sin (w c 



U = u° + 3t, 

y = y Q -J- - COS U 







1 



cos {u° + 3/), 



30 + gjj cos (u<> + 30, 



(2° + y° + — sin u° + — cos u°)e l 



10 



10 



cos U K 



sin (u° 4- St) + — cos (u° + 3f). 

10 v ^ ; ^ 30 v ^ y 

Not even the rate of change of y is constant, yet x and z are 
independent. 

Physically the conclusions are reasonable. The various con- 
ditions which make x and z independent all have the effect of 
lessening or abolishing x's and s's effect on y. The abolition can 
be done either by making y constant, or by driving y exclusively 
by some other variable (u). A well-known example of the latter 
method is the ' jamming ' of a broadcast by the addition of some 

250 



CONSTANCY AND INDEPENDENCE 24/17 

powerful fluctuating signal from another station. It may effec- 
tively render the listener independent of the broadcaster. 

If to the original conditions we add the restriction that the 
system A is to become absolute on being made independent of C, 
then constancy of the variables Xb becomes necessary. For the 
possibilities examined in paragraphs (1) and (2) leave system A 
subject to parameters xr which were assumed to be effective and 
which are now changing. In such conditions A cannot be 
absolute (S. 21/1) : constancy of the variables xn is therefore 
necessary. 

24/16. The statement of S. 14/15, that in an absolute system an 
inactive variable cannot become active unless some variable 
directly affecting it is active, will now be proved. 

Theorem : If a variable x a is related to a set xb so that/ (. . .) 
contains only x a and Xb, and if x a and Xb have all been constant 
over a finite time, and if x a becomes active while the set xb stays 
inactive, then the system cannot be absolute. 

We are given that dx a /dt =f a (x a , x B ). As x a remained con- 
stant (at X a , say) while the set Xb were constant (at Xb), it follows 
that f a (X a , Xb) = 0. But if x a starts to change value, dx a /dt is 
no longer zero, nor is/ a ; sof a (X a , Xb) is a double- valued function 
of its arguments, the system is not state-determined, and it is 
therefore not absolute. 

24/17. In the ' hour-glass ' system of S. 14/11, every variable 
may be shown to be dependent on every other variable. As in 
Figure 14/11/1, let systems A and B each act on, and be acted 
on, by a variable x. The differential matrix, in partitioned form, 
is 





A 


X 


B 


A 


~R 


R 





X 


R 


R 


R 


B 


_0 


R 


R. 



Its square contains only R's. So none of the A's are independent 
of the Z?'s, and conversely. 

The proof is confirmed by the theorem of S. 19/22, which shows 
that, as far as system B is concerned, the values of the A's can 
be replaced by the derivatives of x. The behaviours of all A's 

251 



24/18 



DESIGN FOR A BRAIN 



variables are therefore represented in a?'s behaviour by aj's deriva- 
tives, and Z?'s variables are thus not independent of A's. 



24/18. In S. 14/16 we wanted to compare two probabilities, 
each that a system would be stable, one composed of part- 
functions and the other of full-functions, other things being equal. 
The method of S. 20/12 will define the individual probabilities. 
The question of what we mean by ' other things ' may be treated 
by postulating that, regarded as two random processes, (a) the 
one system's full-functions and (b) the active sections of the other 
system's part-functions are to have the same statistical properties 
(when averaged over all lines of behaviour.) This postulate is 
stated purely in terms of the systems' observable behaviour, so 
that it would be easy, in a given case, to test whether the postu- 
late was satisfied. 

Now consider a system of n variables, part-functions that on 
the average are active over a fraction p of the time. The average 
number of variables active at one time will be pn = k, say. 
Suppose that, at a point Q, the average number of variables are 
active. For convenience, re-label the variables to list the active 
first. Add parameters ol v . . . to generate the distribution. At 
Q we have 

dxjdt = f 1 {x 1 , . . . , x n \ <*!, . . .y 



dx k /dt =f k (x v . . . , x n ; oc v . . .) 

dxk+i/dt = 



dxn/dt = 



The differential matrix at this point will be formally of order 
n X n, but the rows from k + 1 to n will be all zero. If now we 
test the probability of stability at this Q we find that in fact it 
depends on the probability that the a-combination has given (a) 
f x = . . . =fk = 0, and (b) that the matrix 









dfk 
_dx 1 



dx k . 



252 



CONSTANCY AND INDEPENDENCE 24/19 

passes Hurwitz' test. Whatever the probability may be, it is 
clearly equal to the probability of stability given by a system of 
k similar full-functions. 



Conventions and symbols 

24/19. The relations between the various entities defined in this 
chapter are here summarised for convenience of reference. 

(1) (In these four statements take the upper relation in the 
braces in all, or the lower in all). 

(a) xA^ not independent of Xj 

(b) F k {. . . , x] + A4 . . . ; ojljlW • • ' X °P • • ' ' ') ; 

a 



(d) The integral matrix has i -^ j> at the {kj)-th position 
(2) The (hi)-th position is in the h-th row and ^-th column : 

en) (12) (is) . . r 

(21) (22) (23) . . . 



(c)— F k {x° v . . ., x° n ; 0| + 



(3) The differential matrix has <^ j> at (pq) as df p /dx q { i ^0 

„ integral „ „ ,, „ ,, „ dF p /dx q ,, „ 

(4) The following correspond : 

(a) In the diagram of immediate effects : x r — ► x s ; 

(b) In the canonical equations : f 8 (. . . , av, . . .) ; 

(c) In the differential matrix : an J? at the (sr)-th position. 

(5) If sets A and B include all the variables, and if deletion 
from the integral matrix of : 

columns A and rows B leaves all zeros, and 
columns B and rows A leaves not all zeros, 
then A dominates B. 

253 



24/19 






DESIGN FOR 


A 


BRAIN 


(6) 


If x p is a 


part-function 


and 


is 


inactive : 




(a) 


dxp/dt 


= 0; 










(b) 


X p = X 


o . 










M 


Fp(x<> ; 


t) - < ; 









(d) a^/3ajj - (all g + p) ; 

(e) all elements (except (pp) ) of the p-th row of the 

differential and integral matrices are zero. 

Reference 
Riguet, J. Sur les rapports entre les concepts de machine de multipole et de 
structure algebrique. Comptes rendus des seances de V Academic des 
Sciences, 237, 425 ; 1953. 



254 



INDEX 

(The number refers to the page. A bold-faced number indicates a definition.) 



Absolute system, 24, 45, 73, 105, 

209, Chapter 19 
Accumulation, 213 
Activity, 67, 162, 164 
Adaptation, 64, Chapter 5 

accumulation of, 172 

loss of, 133 

needs independence, 137 

of iterated systems, 141 

of multistable system, 172 

serial, Chapter 17 

system to system, 174 

time taken, 135 

two meanings, 63 
Adaptive behaviour, 64 

classified by Holmes, 66 
Addition of stimuli, 167 
Aileron, 99 
Aim : see Goal 
Algebra, of machine, 254 
All or nothing, 127 
Ammonia, 78 
Ammonium chloride, 20 
Amoeba, 152 

Amoeboid movement, 126 
Animal-centred co-ordinates, 40 
Archimedes, 65 
Area striata, 170 
Artificial limb, 39 
Ashby, W. Ross, 71, 102, 199, 225, 

231, 236, 240 
Assembly, and canonical equations, 

211 
Association areas, 190 
Assumptions made, 9 
Automatic pilot, 99 
Awareness, 10 

Bacteria, 156 
Bartlett, F. S., 37, 42 
Behaviour, 14 

classified, 66 

reflex and learned, 2 

representation of, 23 
Bernard, Claude, 125 
Bicycle, 35 
Bigelow, J., 71 
Binocular vision, 118 
Biochemical co-ordination, 196 
Body-temperature, 58 
Borrowed knowledge, 14 
Boyd, D. A., 131, 138 



Break, 85, 126, 233 
Burn, 131 

Calculating machine, 130 
Cannon, W. B., 57, 64, 71 
Canonical equations, 206, 211 
Cardinal number, 31 
Carey, E. J., 127, 129 
Carmine, 103 
Causation, 49 

partition of, 241 
Cell, step-functions in, 125 
Characteristic equation, 217 
Chemical dynamics, 17, 143, 178 
Chess, 8, 102 

Chewing, co-ordination of, 7 
Chimney, 191 
Chimpanzee, 186 
Civilisation, 62 
Clock, as time-indicator, 15 

variables of, 14 
Cochlea, 169 

Collision, 60, 131, 180, 187 
Complexity, 121 
Compound microscope, 3 
Concepts, restrictions on, 9 
Conditioned reflex, 2, 8, 16, 19, 34, 75, 

115, 137, 167, 192, 194 
Conduction of heat, 17 
Congruence, 205 

Connection between systems, 153 
Consciousness, 10 

Conservation, of adaptation, 133, 140, 
184 

of zeros, 231 
Constancy, 67, 72, Chapter 14 

classification of, 80 
Constant, as null-function, 203 

of proportionality, 81, 84, 232 
Constraint, adaptation to, 102 
Control, 16, 155 

"by error, 54 
Convulsion, 187 
Co-ordinates, animal-centred, 40 

change of, 212 
Co-ordination, and stability, 55 

motor, 67 

of reflexes, 196 
Cortex, localisation in, 191 
Cough reflex, 2 
Cowles, J. T., 186, 189 
Critical state, 84 



255 



INDEX 



Critical state and goal, 120 

distribution of, 92 

necessity of, 112 

of endrome, 128 
Critical surface, 234 

and essential variables, 130 
Crystallisation, 143 
Cube, equilibrium of, 45 
Culler, E., 75, 79, 115, 123, 186, 189 
Curare, 75 
Cybernetics, 154 
Cycle, stable, 48 
Cycling, 10, 35 

DAMS, 171 

Delay, between trials, 132 

in canonical equations, 213 
Delicacy, of neuron, 126 
Demonstrability, 9, 11 
Dependence, 155 

Derivatives as variables, 162, 212 
Determinateness, 10, 111 
Diabetes, 75 

Diagram of immediate effects, 50, 157 
Diagram of ultimate effects, 158, 160 
Dial-readings, 14, 29 
Differential equations, 206 
Differential matrix, 245 
Digits, terminal, 142 
Dirac's 8-function, 233 
Discontinuity, in environment, 131 
Discrimination, 167 
Dispersion, 166, Chapter 15 

and ultrastability, 176 

in multistable system, 172 

in sense organs, 169 

of new learning, 194 
Displacement, 145 
Disturbance, 239, Chapter 13 
Dominance, 158, 185, 246 
Ducklings, 38 
Duncan, C. P., 187, 189 
Dynamic systems, Chapters 2 and 19 

Eddington, A. S., 15, 28 
Effect, 49, 241 
Effector, 120 
Elastic, 81, 232 
Endrome, 128 

Energy, 5, 43, 154, 164, 215 
Engine driver, 6 
Environment, 35 

and homeostasis, 60 

discontinuous, 131 

functional criterion, 118 

iterated, 139 

nature of, 179 

number of variables, 136 

scale of difficulty, 132 

types of, 183 



Epistemology, 211 
Equation, canonical, 206 

characteristic, 217 
Equilibrium, 43 
Error, control by, 54 

correction of, 54 
Essential variable, 41 

and adaptation, 64 

and critical surfaces, 122, 130 

and normal equilibrium, 146 

and Stentor, 105 
Evolution, 8, 196 
Experience, 119 
Experiment, structure of, 31 
Experimenter, during training, 113 
Eyeball, 117 

Falcon, training of, 186 
Fatigue, and habituation, 152 
Feedback, 51 

and stability, 52 

demonstration of, 49 

in homeostat, 95 

in neuronic circuits, 128 

in physiology, 37 

in training, 114 

organism-environment, 36 
Feeding, 113 
Fencer, 67 
Fibrils, neuro-, 126 
Field, 22 

destroyed, 145 

multiple, 88, 235 

of absolute system, 26 

of regular system, 210 

parameter-change, 74 

part-functions, 163 

stability of, 84 
Finite continuous group, 205 
Fire, 3 

Fisher, R. A., 96, 102, 222 
Fit, 187 

Fixing a variable, 55, 220 
Fracture, 131 
Frazer, R. A., 225 
Freezing of spinal cord, 160 
Full-function, 80 

if ignored, 86 
Function-rules, 8, 122 
Fuse, as step-function, 81 

critical state, 84 

Gene-pattern, 8, 122, 197 

Gestalt-recognition, 168 

Gestalt school, 137 

Girden, E., 75, 79 

Glucose in blood, and diabetes, 75 

homeostasis of, 58 
Goal-seeking, 53 

control of aim, 120 



256 



INDEX 



Goal-seeking, inappropriate, 130 
Governor, see Watt's governor 
Gradation, in homeostat, 133 
in iterated systems, 140 
in multistable system, 184 
Grant, W. T., G8, 71 
Grindley, G. C, 114, 124 
Group, equations of, 204 
of equivalent patterns, 108 

Habituation, 151 

intracerebral, 195 
Haemorrhage, 33 
Harmonic oscillator, 80 
Harrison, R. G., 120, 129 
Hawking, 180 

Hilgard, E. R., 115, 124, 194 
Holmes, S. J., 00, 71 
Homeostasis (Cannon), 57 
Homeostat, 

adapting, 109 

construction, 93 

delay between trials, 132 

difficulty of stabilisation, 105 

equations of, 207, 220 

habituation, 148 

interaction, 175 

modes of failure, 130 

trained, 110 

two environments, 134 
Hormone, 122 
Hour-glass system, 101, 251 
House, for homeostasis, 02 
Humphrey, G., 152 
Hunger, 58 
Hunting, 132 
Hurwitz, A., 225 
Hurwitz' test, 218, 223 

Idempotency, 244 
Immediate effect, 50 
Immunity to displacement, 140 
Inactivity, 162 

Independence, 155, Chapters 14 
24 

not arbitrary, 157 

ultimate effects, 158 
Individuality of subsystems, 170 
Inflammation, 9 
Information, 154, 211 
Initial state, 19 

control over, 10, 78 
Insensitivity, 131 
Instability, see Stability 
Instrumental lean 
Insulator, 153, 15* 
Integral matrix, 245 
Integration, 212, 214 
Intelligence test, 132 
Interaction, 174, Chapter 18 



and 



Interference, Principle of, 194 

Intrinsic stability, 219 

Invariant, 108, 231 

Isolation, 153, 159, 210 

Iterated systems, 140, 184, Chapter 12 

Jacobian, 210 

Jamming, 250 

Jennings, H. S., 37, 42, 103, 152, 180 

Joining systems, 55, 229 

Kitten, 3, 11, 37, 01, 90, 195 
Krebs' cycle, 190 

Lashley, K. S., 192, 193, 199 

Latent roots, 218 

Law of Reciprocity of Connections, 

128 
Learning 

and adaptation, 03 

and consciousness, 10 

and ultrastable system, 119 

effect of new, 193 

habituation as, 152 

irreversible, 187 

localisation of, 140, 190 

serial adaptation, 180 
Lens, 170 

Lethal environment, 131 
Levi, G., 120, 129 
Liddell, H. S., 19, 28 
Lie group, 205 
Life, 29 
Line of behaviour, 19 

equality of two, 19 

in absolute system, 20, 208 

recording, 19 

of part-functions, 104 

stability of, 47 
Linear system, 215 
Localisation of learning, 190 
Locomotion, 40 
Loeb, Jacques, 120 
Lorente de X6, R., 127, 129 

Machine, 13 

algebra of, 254 

number of variables, 15 
Main variables, 87, 228 
Marina, A., 117, 124 
Mathematics, learning, 185 
Matrix, 

differential, 245 

integral, 245 

notation, 215 

R0— , 243 
Marquis, D. G., 115, 124, 194 
Maze, 3 

McCulloch, W. S., 128, 129, 108, 170 
McDougall, W., 04, 71 



257 



INDEX 



Mechanical brain, 130, 179 

Memory, 119 

permanence of, 190 

Method, 15 

Metrazol, 127 

Microscope, compound, 3 

Micturition, 84 

Minnows, 113 

Morgan, C. Lloyd, 38, 42, 180 

Motor-car, and homeostasis, 62 

Motor co-ordination, see Co-ordina- 
tion 

Motor end-plate, 127 

Mowrer, O. H., 106, 124 

Miiller, G. E., 194, 199 

Multistable system, 171, Chapters 
16-18 
necessity of, 182 

Natural selection, 123, 144, 197 

Natural system, 23 

Nervous system, assumptions, 9, 34, 
190 

Neuron, 5 

number of, in Man, 8 

Neuronic circuit, 128 

Neutral equilibrium, 43 

Noise, 211 

Normal equilibrium, 146, 216, 238 

Null-function, 81, 227 
and absolute system, 86 
separates systems, 159, 173 

Number, 29 

Nyquist, H., 225 

Nyquist's test, 219 

Objectivity, 9 
Observation, 15 
Obstacle, 64, 69 
Oesophageal fistula, 113 
Order of time-scale, 82 
Organisation, 7, 70, 110 
Organism and environment, 35 
Ovum, 122 

Pain, insensitivity to, 131 
Paramecium, 

habituates, 152 

environment of, 180 
Parameter, 72, Chapters 6 and 21 

alternation of, 134, 147 

and disturbance, 145 

and goal, 121 

compound, 32 

effective, 73, 153, 156 

generating step-function, 83 

reaction to, 101 

stabilisation by, 165 
Parker, G., 140, 143, 156 
Part-function, 80 



Part-function and dispersion, 166 

and independence, 247 

and stabilisation, 165 

examples, 162 

in environment, 180 

in multistable system, 171 
Parts, relation to whole, 5, 211 
Pattern of reaction, 33 
Pattern-recognition, 168 
Patterned zeros, 244 
Pavlov, I. P., 16, 64, 71, 115, 167, 

192, 194 
Pendulum, 15 

as absolute system, 25 

field of, 25 

is goal-seeking, 53 

parameters to, 72, 86 
Perception, 11 
Phase-space, 20, 203 

step-function in, 87 

part-function in, 164 
Pike, 113, 131 
Pitts, W., 168, 170 
Poison, 130, 132 
Predictor, 154 
Primary operation, 17, 203 

and field, 23 

and independence, 153, 157 

and line of behaviour, 19 

and stability, 49 
Principle of Interference, 194 

of Ultrastability, 91 
Problem, stated, 11, 70, 90 
Processes, fast and slow, 177 
Progression, in adaptation, 141 
Protoplasm, 126 
Protozoa, habituation in, 152 
Psychological concepts, 9 
Punishment, 112 

Random numbers, 96, 222 
Reaction, in radio, 51 
Receptor, control of aim, 120 

dispersion, 169 
Reducible system, 247 
Reflexes, 2 

co-ordination of, 196 

independence of, 137 
Region of stability, 47 
Regular system, 23, 204, 209 
Rein, H., 33, 42 
Representative point, 20 
Response, to stimulus, 166 

to repeated stimuli, 147 
Resting state, 48, 216 
Reverberating circuits, 128 
Reversal, reaction to, 98 
Reward, 112 
Riguet, J., 242, 254 
Rigidity, 52 



258 



INDEX 



JBO-matrix, 243 
Robinson, E. S., 194, 199 
Rosenblueth, A., 71 
Runaway, 52 

Sea- anemone, 140, 156 
Selection of fields, 91, 106 
Self-correction, 54 
Self-destruction, 5 
Sense-organs, dispersion in, 169 
Separation, 153, 160 
Serum, 20 

Servo-mechanism, 51 
Sex hormone, 122 
Shannon, C. E., 211, 215 
Sherrington, C. S., 137, 138 
Shivering, 58 

Simple harmonic oscillator, 80 
Skaggs, E. B., 194, 199 
Skin, 169 

Skinner, B. F., 190, 199 
Snail, 152 
Snake, 135 
Sommerhoff, A., 71 
Species, 8 
Spectrum, 30 
Speidel, C. C, 127, 129 
Sperry, R. W., 117, 124 
Spider monkey, 118 
Spinal cord, transection, 159 
Spinal reflexes, co-ordination between, 
196 

interaction between, 137 
Spontaneity, 88 
Stability, 47, Chapters 4 and 20 

and parameter change, 78 

examples, 55 

is holistic, 54 

nature of, 76 

of homeostat, 96, 220 

probability of, 56, 100, 136, 165, 221 
Stalking, 132 

Starling, E. H., 37, 42, 64 
State, 18, 203 

equality of two, 19 
State-determined system, 25 
Statistical machinery, 145 
Steady state, 43 
Stentor, 103 
Step-function, 80, Chapters 7 and 22 

and natural selection, 123 

effect of omission, 88 

in Stentor, 105 

in the organism, Chapter 10 

nature of, 111 

necessity of, 110 

number necessary, 128 
Stepping switch, 96 
Stimuli, 

addition of, 167 



Stimuli, and part-functions, 166 

numericising, 30, 76 

repeated, 147 

response to, 57 
Strychnine, 130 
Subjective phenomena, 10, 32 
Subspaces, 87, 163 
Subsystems, 171, 181 
Surface, critical, 234 
Surgical alterations, 117 
Survival, 42, 65, 197 
Swallow, 17 

Switch, as step-function, 81 
Switching, and part-function, 161 
Symbols, collected, 253 
System, 15 

absolute, 24, Chapter 19 

containing null-functions, 86, 228 

— part-functions, 163 

— step-functions, 86 
hour-glass, 161, 251 
independence of, 156 
infinite, 203 
linear, 215 
regular, 23, 204 
symbols for, 203 
stability of, 48 
with feedback, 39 

Tabes dorsalis, 38 
Tadpoles, 127 
Taste, 169 
Teleology, 9, 71 
Telephone exchange, 161 
Temperature, homeostasis of, 58 
Temple, G., 27, 28 
Temporal cortex, 170 
Terminal field, 91 

normality, 146 
Thermostat, 43, 51, 53, 54, 67, 156 
Thirst, 59 

Threshold, 163, 170 
Thunderstorm, 17 
Time, 

as variable, 15 

for adaptation, 135, 141, 177, 184 
Tissue culture, 126 
Tokens, as reward, 186 
Tongue, 7, 39 
Training, 112, 187 
Transducer, 121 

noiseless, 211 
Transformations, 168 
Transient, 144 
Trap, 132 
Tremor, 67 

Trial and error, 112, 132, 174 
Trials, number of, 238 
Type-problem, 11 
Typewriter, 154 



259 



INDEX 



Ultrastability 

and dispersion, 171, 17G 

and habituation, 152 

and interaction, 195 
Ultrastable system, 91, Chapters 8 
and 23 

specified by genes, 122 

formed by natural selection, 123 
Uniselector, 96 
Universals, 170 
Unstable equilibrium, 43 
Urinary bladder, 84 

Variable, 14 

and disturbance, 145 
independence, 15G 
replacement by derivative, 212 
restricted meaning, 72 



Velocity, 214 
Vicious circle, 52 
Visual cortex, 170 
Vital properties, 9 

Walking, 38 

Water, homeostasis of, 59 

Watt's governor, 43, 46, 49, 154 

Weather, 32 

Wholeness 

of nervous system, 137 

of environment, 184 
Wiener, N., 56, 71, 154 
Wolfe, J. B., 186, 189 

Yates, F., 96, 102 

Zeros, patterned, 244